Subtract (5x)/(2x^2+5x-3)-(2x)/(x^2+6x+9)

$\frac{5 x}{2 x^{2} + 5 x - 3} - \frac{2 x}{x^{2} + 6 x + 9}$

Simplify each term.

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Factor by grouping.

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For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = 5$ .

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Factor $5$ out of $5 x$ .

$\frac{5 x}{2 x^{2} + 5 (x) - 3} - \frac{2 x}{x^{2} + 6 x + 9}$

Rewrite $5$ as $- 1$ plus $6$

$\frac{5 x}{2 x^{2} + (- 1 + 6) x - 3} - \frac{2 x}{x^{2} + 6 x + 9}$

Apply the distributive property.

$\frac{5 x}{2 x^{2} - 1 x + 6 x - 3} - \frac{2 x}{x^{2} + 6 x + 9}$

Factor out the greatest common factor from each group.

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Group the first two terms and the last two terms.

$\frac{5 x}{(2 x^{2} - 1 x) + 6 x - 3} - \frac{2 x}{x^{2} + 6 x + 9}$

Factor out the greatest common factor (GCF) from each group.

$\frac{5 x}{x (2 x - 1) + 3 (2 x - 1)} - \frac{2 x}{x^{2} + 6 x + 9}$

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$\frac{5 x}{(2 x - 1) (x + 3)} - \frac{2 x}{x^{2} + 6 x + 9}$

Factor using the perfect square rule.

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Rewrite $9$ as $3^{2}$ .

$\frac{5 x}{(2 x - 1) (x + 3)} - \frac{2 x}{x^{2} + 6 x + 3^{2}}$

Check the middle term by multiplying $2 a b$ and compare this result with the middle term in the original expression.

$2 a b = 2 \cdot x \cdot 3$

Simplify.

$2 a b = 6 x$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 3$ .

$\frac{5 x}{(2 x - 1) (x + 3)} - \frac{2 x}{{(x + 3)}^{2}}$

To write $\frac{5 x}{(2 x - 1) (x + 3)}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$\frac{5 x}{(2 x - 1) (x + 3)} \cdot \frac{x + 3}{x + 3} - \frac{2 x}{{(x + 3)}^{2}}$

To write $- \frac{2 x}{{(x + 3)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{2 x - 1}{2 x - 1}$ .

$\frac{5 x}{(2 x - 1) (x + 3)} \cdot \frac{x + 3}{x + 3} - \frac{2 x}{{(x + 3)}^{2}} \cdot \frac{2 x - 1}{2 x - 1}$

Write each expression with a common denominator of $(2 x - 1) {(x + 3)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{5 x}{(2 x - 1) (x + 3)}$ and $\frac{x + 3}{x + 3}$ .

$\frac{5 x (x + 3)}{(2 x - 1) (x + 3) (x + 3)} - \frac{2 x}{{(x + 3)}^{2}} \cdot \frac{2 x - 1}{2 x - 1}$

Raise $x + 3$ to the power of $1$ .

$\frac{5 x (x + 3)}{(2 x - 1) ({(x + 3)}^{1} (x + 3))} - \frac{2 x}{{(x + 3)}^{2}} \cdot \frac{2 x - 1}{2 x - 1}$

Raise $x + 3$ to the power of $1$ .

$\frac{5 x (x + 3)}{(2 x - 1) ({(x + 3)}^{1} {(x + 3)}^{1})} - \frac{2 x}{{(x + 3)}^{2}} \cdot \frac{2 x - 1}{2 x - 1}$

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

$\frac{5 x (x + 3)}{(2 x - 1) {(x + 3)}^{1 + 1}} - \frac{2 x}{{(x + 3)}^{2}} \cdot \frac{2 x - 1}{2 x - 1}$

Add $1$ and $1$ .

$\frac{5 x (x + 3)}{(2 x - 1) {(x + 3)}^{2}} - \frac{2 x}{{(x + 3)}^{2}} \cdot \frac{2 x - 1}{2 x - 1}$

Multiply $\frac{2 x}{{(x + 3)}^{2}}$ and $\frac{2 x - 1}{2 x - 1}$ .

$\frac{5 x (x + 3)}{(2 x - 1) {(x + 3)}^{2}} - \frac{2 x (2 x - 1)}{{(x + 3)}^{2} (2 x - 1)}$

Reorder the factors of $(2 x - 1) {(x + 3)}^{2}$ .

$\frac{5 x (x + 3)}{{(x + 3)}^{2} (2 x - 1)} - \frac{2 x (2 x - 1)}{{(x + 3)}^{2} (2 x - 1)}$

Combine the numerators over the common denominator.

$\frac{5 x (x + 3) - 2 x (2 x - 1)}{{(x + 3)}^{2} (2 x - 1)}$

Simplify the numerator.

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Factor $x$ out of $5 x (x + 3) - 2 x (2 x - 1)$ .

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Factor $x$ out of $5 x (x + 3)$ .

$\frac{x (5 (x + 3)) - 2 x (2 x - 1)}{{(x + 3)}^{2} (2 x - 1)}$

Factor $x$ out of $- 2 x (2 x - 1)$ .

$\frac{x (5 (x + 3)) + x (- 2 (2 x - 1))}{{(x + 3)}^{2} (2 x - 1)}$

Factor $x$ out of $x (5 (x + 3)) + x (- 2 (2 x - 1))$ .

$\frac{x (5 (x + 3) - 2 (2 x - 1))}{{(x + 3)}^{2} (2 x - 1)}$

Apply the distributive property.

$\frac{x (5 x + 5 \cdot 3 - 2 (2 x - 1))}{{(x + 3)}^{2} (2 x - 1)}$

Multiply $5$ by $3$ .

$\frac{x (5 x + 15 - 2 (2 x - 1))}{{(x + 3)}^{2} (2 x - 1)}$

Apply the distributive property.

$\frac{x (5 x + 15 - 2 (2 x) - 2 \cdot - 1)}{{(x + 3)}^{2} (2 x - 1)}$

Multiply $2$ by $- 2$ .

$\frac{x (5 x + 15 - 4 x - 2 \cdot - 1)}{{(x + 3)}^{2} (2 x - 1)}$

Multiply $- 2$ by $- 1$ .

$\frac{x (5 x + 15 - 4 x + 2)}{{(x + 3)}^{2} (2 x - 1)}$

Subtract $4 x$ from $5 x$ .

$\frac{x (x + 15 + 2)}{{(x + 3)}^{2} (2 x - 1)}$

Add $15$ and $2$ .

$\frac{x (x + 17)}{{(x + 3)}^{2} (2 x - 1)}$

Subtract (5x)/(2x^2+5x-3)-(2x)/(x^2+6x+9)

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