Evaluate h(x)=x^2+x+2 , (h(3+deltax)-h*3)/(deltax)

$h (x) = x^{2} + x + 2$ , $\frac{h (3 + d e l t a x) - h \cdot 3}{d e l t a x}$

Evaluate the numerator in $\frac{h (3 + d e l t a x) - h \cdot 3}{d e l t a x}$ .

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Replace the variable $x$ with $3 + d e l t a x$ in the expression.

$h (3 + d e l t a x) = {(3 + d e l t a x)}^{2} + 3 + d e l t a x + 2$

Replace the function designators in $h (3 + d e l t a x) - h \cdot 3$ with the actual functions.

${(3 + d e l t a x)}^{2} + 3 + d e l t a x + 2 - (x^{2} + x + 2) \cdot 3$

Simplify ${(3 + d e l t a x)}^{2} + 3 + d e l t a x + 2 - (x^{2} + x + 2) \cdot 3$ .

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Remove parentheses.

${(3 + d e l t a x)}^{2} + 3 + d e l t a x + 2 - (x^{2} + x + 2) \cdot 3$

Simplify each term.

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Rewrite ${(3 + d e l t a x)}^{2}$ as $(3 + d e l t a x) (3 + d e l t a x)$ .

$(3 + d e l t a x) (3 + d e l t a x) + 3 + d e l t a x + 2 - (x^{2} + x + 2) \cdot 3$

Expand $(3 + d e l t a x) (3 + d e l t a x)$ using the FOIL Method.

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Apply the distributive property.

$3 (3 + d e l t a x) + d e l t a x (3 + d e l t a x) + 3 + d e l t a x + 2 - (x^{2} + x + 2) \cdot 3$

Apply the distributive property.

$3 \cdot 3 + 3 (d e l t a x) + d e l t a x (3 + d e l t a x) + 3 + d e l t a x + 2 - (x^{2} + x + 2) \cdot 3$

Apply the distributive property.

$3 \cdot 3 + 3 (d e l t a x) + d e l t a x \cdot 3 + d e l t a x (d e l t a x) + 3 + d e l t a x + 2 - (x^{2} + x + 2) \cdot 3$

Simplify and combine like terms.

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Simplify each term.

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Multiply $3$ by $3$ .

$9 + 3 (d e l t a x) + d e l t a x \cdot 3 + d e l t a x (d e l t a x) + 3 + d e l t a x + 2 - (x^{2} + x + 2) \cdot 3$

Move $3$ to the left of $d e l t a x$ .

$9 + 3 d e l t a x + 3 \cdot (d e l t a x) + d e l t a x (d e l t a x) + 3 + d e l t a x + 2 - (x^{2} + x + 2) \cdot 3$

Multiply $d$ by $d$ by adding the exponents.

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Move $d$ .

$9 + 3 d e l t a x + 3 d e l t a x + d \cdot d e l t a x (e l t a x) + 3 + d e l t a x + 2 - (x^{2} + x + 2) \cdot 3$

Multiply $d$ by $d$ .

$9 + 3 d e l t a x + 3 d e l t a x + d^{2} e l t a x (e l t a x) + 3 + d e l t a x + 2 - (x^{2} + x + 2) \cdot 3$

Multiply $l$ by $l$ by adding the exponents.

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Move $l$ .

$9 + 3 d e l t a x + 3 d e l t a x + d^{2} e (l \cdot l) t a x (e t a x) + 3 + d e l t a x + 2 - (x^{2} + x + 2) \cdot 3$

Multiply $l$ by $l$ .

$9 + 3 d e l t a x + 3 d e l t a x + d^{2} e l^{2} t a x (e t a x) + 3 + d e l t a x + 2 - (x^{2} + x + 2) \cdot 3$

Multiply $t$ by $t$ by adding the exponents.

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Move $t$ .

$9 + 3 d e l t a x + 3 d e l t a x + d^{2} e l^{2} (t \cdot t) a x (e a x) + 3 + d e l t a x + 2 - (x^{2} + x + 2) \cdot 3$

Multiply $t$ by $t$ .

$9 + 3 d e l t a x + 3 d e l t a x + d^{2} e l^{2} t^{2} a x (e a x) + 3 + d e l t a x + 2 - (x^{2} + x + 2) \cdot 3$

Multiply $a$ by $a$ by adding the exponents.

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Move $a$ .

$9 + 3 d e l t a x + 3 d e l t a x + d^{2} e l^{2} t^{2} (a \cdot a) x (e x) + 3 + d e l t a x + 2 - (x^{2} + x + 2) \cdot 3$

Multiply $a$ by $a$ .

$9 + 3 d e l t a x + 3 d e l t a x + d^{2} e l^{2} t^{2} a^{2} x (e x) + 3 + d e l t a x + 2 - (x^{2} + x + 2) \cdot 3$

Multiply $x$ by $x$ by adding the exponents.

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Move $x$ .

$9 + 3 d e l t a x + 3 d e l t a x + d^{2} e l^{2} t^{2} a^{2} (x \cdot x) e + 3 + d e l t a x + 2 - (x^{2} + x + 2) \cdot 3$

Multiply $x$ by $x$ .

$9 + 3 d e l t a x + 3 d e l t a x + d^{2} e l^{2} t^{2} a^{2} x^{2} e + 3 + d e l t a x + 2 - (x^{2} + x + 2) \cdot 3$

Multiply $d^{2} e l^{2} t^{2} a^{2} x^{2} e$ .

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Raise $e$ to the power of $1$ .

$9 + 3 d e l t a x + 3 d e l t a x + d^{2} l^{2} t^{2} a^{2} x^{2} (e^{1} e) + 3 + d e l t a x + 2 - (x^{2} + x + 2) \cdot 3$

Raise $e$ to the power of $1$ .

$9 + 3 d e l t a x + 3 d e l t a x + d^{2} l^{2} t^{2} a^{2} x^{2} (e^{1} e^{1}) + 3 + d e l t a x + 2 - (x^{2} + x + 2) \cdot 3$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$9 + 3 d e l t a x + 3 d e l t a x + d^{2} l^{2} t^{2} a^{2} x^{2} e^{1 + 1} + 3 + d e l t a x + 2 - (x^{2} + x + 2) \cdot 3$

Add $1$ and $1$ .

$9 + 3 d e l t a x + 3 d e l t a x + d^{2} l^{2} t^{2} a^{2} x^{2} e^{2} + 3 + d e l t a x + 2 - (x^{2} + x + 2) \cdot 3$

Add $3 d e l t a x$ and $3 d e l t a x$ .

$9 + 6 d e l t a x + d^{2} l^{2} t^{2} a^{2} x^{2} e^{2} + 3 + d e l t a x + 2 - (x^{2} + x + 2) \cdot 3$

Apply the distributive property.

$9 + 6 d e l t a x + d^{2} l^{2} t^{2} a^{2} x^{2} e^{2} + 3 + d e l t a x + 2 + (- x^{2} - x - 1 \cdot 2) \cdot 3$

Multiply $- 1$ by $2$ .

$9 + 6 d e l t a x + d^{2} l^{2} t^{2} a^{2} x^{2} e^{2} + 3 + d e l t a x + 2 + (- x^{2} - x - 2) \cdot 3$

Apply the distributive property.

$9 + 6 d e l t a x + d^{2} l^{2} t^{2} a^{2} x^{2} e^{2} + 3 + d e l t a x + 2 - x^{2} \cdot 3 - x \cdot 3 - 2 \cdot 3$

Simplify.

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Multiply $3$ by $- 1$ .

$9 + 6 d e l t a x + d^{2} l^{2} t^{2} a^{2} x^{2} e^{2} + 3 + d e l t a x + 2 - 3 x^{2} - x \cdot 3 - 2 \cdot 3$

Multiply $3$ by $- 1$ .

$9 + 6 d e l t a x + d^{2} l^{2} t^{2} a^{2} x^{2} e^{2} + 3 + d e l t a x + 2 - 3 x^{2} - 3 x - 2 \cdot 3$

Multiply $- 2$ by $3$ .

$9 + 6 d e l t a x + d^{2} l^{2} t^{2} a^{2} x^{2} e^{2} + 3 + d e l t a x + 2 - 3 x^{2} - 3 x - 6$

Simplify by adding terms.

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Add $9$ and $3$ .

$6 d e l t a x + d^{2} l^{2} t^{2} a^{2} x^{2} e^{2} + 12 + d e l t a x + 2 - 3 x^{2} - 3 x - 6$

Add $6 d e l t a x$ and $d e l t a x$ .

$7 d e l t a x + d^{2} l^{2} t^{2} a^{2} x^{2} e^{2} + 12 + 2 - 3 x^{2} - 3 x - 6$

Simplify the expression.

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Add $12$ and $2$ .

$7 d e l t a x + d^{2} l^{2} t^{2} a^{2} x^{2} e^{2} + 14 - 3 x^{2} - 3 x - 6$

Subtract $6$ from $14$ .

$7 d e l t a x + d^{2} l^{2} t^{2} a^{2} x^{2} e^{2} - 3 x^{2} - 3 x + 8$

Move $e$ .

$7 d l t a x e + d^{2} l^{2} t^{2} a^{2} x^{2} e^{2} - 3 x^{2} - 3 x + 8$

Move $t$ .

$7 d l a t x e + d^{2} l^{2} t^{2} a^{2} x^{2} e^{2} - 3 x^{2} - 3 x + 8$

Move $l$ .

$7 d a l t x e + d^{2} l^{2} t^{2} a^{2} x^{2} e^{2} - 3 x^{2} - 3 x + 8$

Move $d$ .

$7 a d l t x e + d^{2} l^{2} t^{2} a^{2} x^{2} e^{2} - 3 x^{2} - 3 x + 8$

Move $t^{2}$ .

$7 a d l t x e + d^{2} l^{2} a^{2} t^{2} x^{2} e^{2} - 3 x^{2} - 3 x + 8$

Move $l^{2}$ .

$7 a d l t x e + d^{2} a^{2} l^{2} t^{2} x^{2} e^{2} - 3 x^{2} - 3 x + 8$

Reorder $d^{2}$ and $a^{2}$ .

$7 a d l t x e + a^{2} d^{2} l^{2} t^{2} x^{2} e^{2} - 3 x^{2} - 3 x + 8$

Reorder $7 a d l t x e$ and $a^{2} d^{2} l^{2} t^{2} x^{2} e^{2}$ .

$a^{2} d^{2} l^{2} t^{2} x^{2} e^{2} + 7 a d l t x e - 3 x^{2} - 3 x + 8$

Rewrite $\frac{h (3 + d e l t a x) - h \cdot 3}{d e l t a x}$ .

$\frac{a^{2} d^{2} l^{2} t^{2} x^{2} e^{2} + 7 a d l t x e - 3 x^{2} - 3 x + 8}{d e l t a x}$

Evaluate h(x)=x^2+x+2 , (h(3+deltax)-h*3)/(deltax)

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