Add (2y)/(y^2-2y+1)+8/(y^2+1y-2)

$\frac{2 y}{y^{2} - 2 y + 1} + \frac{8}{y^{2} + 1 y - 2}$

Simplify each term.

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Factor using the perfect square rule.

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Rewrite $1$ as $1^{2}$ .

$\frac{2 y}{y^{2} - 2 y + 1^{2}} + \frac{8}{y^{2} + 1 y - 2}$

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

$2 a b = 2 \cdot y \cdot - 1$

Simplify.

$2 a b = - 2 y$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = y$ and $b = - 1$ .

$\frac{2 y}{{(y - 1)}^{2}} + \frac{8}{y^{2} + 1 y - 2}$

Factor $y^{2} + 1 y - 2$ using the AC method.

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Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $1$ .

$- 1, 2$

Write the factored form using these integers.

$\frac{2 y}{{(y - 1)}^{2}} + \frac{8}{(y - 1) (y + 2)}$

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

$\frac{2 y}{{(y - 1)}^{2}} \cdot \frac{y + 2}{y + 2} + \frac{8}{(y - 1) (y + 2)}$

To write $\frac{8}{(y - 1) (y + 2)}$ as a fraction with a common denominator, multiply by $\frac{y - 1}{y - 1}$ .

$\frac{2 y}{{(y - 1)}^{2}} \cdot \frac{y + 2}{y + 2} + \frac{8}{(y - 1) (y + 2)} \cdot \frac{y - 1}{y - 1}$

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

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

$\frac{2 y (y + 2)}{{(y - 1)}^{2} (y + 2)} + \frac{8}{(y - 1) (y + 2)} \cdot \frac{y - 1}{y - 1}$

Multiply $\frac{8}{(y - 1) (y + 2)}$ and $\frac{y - 1}{y - 1}$ .

$\frac{2 y (y + 2)}{{(y - 1)}^{2} (y + 2)} + \frac{8 (y - 1)}{(y - 1) (y + 2) (y - 1)}$

Raise $y - 1$ to the power of $1$ .

$\frac{2 y (y + 2)}{{(y - 1)}^{2} (y + 2)} + \frac{8 (y - 1)}{{(y - 1)}^{1} (y - 1) (y + 2)}$

Raise $y - 1$ to the power of $1$ .

$\frac{2 y (y + 2)}{{(y - 1)}^{2} (y + 2)} + \frac{8 (y - 1)}{{(y - 1)}^{1} {(y - 1)}^{1} (y + 2)}$

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

$\frac{2 y (y + 2)}{{(y - 1)}^{2} (y + 2)} + \frac{8 (y - 1)}{{(y - 1)}^{1 + 1} (y + 2)}$

Add $1$ and $1$ .

$\frac{2 y (y + 2)}{{(y - 1)}^{2} (y + 2)} + \frac{8 (y - 1)}{{(y - 1)}^{2} (y + 2)}$

Combine the numerators over the common denominator.

$\frac{2 y (y + 2) + 8 (y - 1)}{{(y - 1)}^{2} (y + 2)}$

Simplify the numerator.

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Factor $2$ out of $2 y (y + 2) + 8 (y - 1)$ .

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Factor $2$ out of $2 y (y + 2)$ .

$\frac{2 (y (y + 2)) + 8 (y - 1)}{{(y - 1)}^{2} (y + 2)}$

Factor $2$ out of $8 (y - 1)$ .

$\frac{2 (y (y + 2)) + 2 (4 (y - 1))}{{(y - 1)}^{2} (y + 2)}$

Factor $2$ out of $2 (y (y + 2)) + 2 (4 (y - 1))$ .

$\frac{2 (y (y + 2) + 4 (y - 1))}{{(y - 1)}^{2} (y + 2)}$

Apply the distributive property.

$\frac{2 (y \cdot y + y \cdot 2 + 4 (y - 1))}{{(y - 1)}^{2} (y + 2)}$

Multiply $y$ by $y$ .

$\frac{2 (y^{2} + y \cdot 2 + 4 (y - 1))}{{(y - 1)}^{2} (y + 2)}$

Move $2$ to the left of $y$ .

$\frac{2 (y^{2} + 2 \cdot y + 4 (y - 1))}{{(y - 1)}^{2} (y + 2)}$

Apply the distributive property.

$\frac{2 (y^{2} + 2 y + 4 y + 4 \cdot - 1)}{{(y - 1)}^{2} (y + 2)}$

Multiply $4$ by $- 1$ .

$\frac{2 (y^{2} + 2 y + 4 y - 4)}{{(y - 1)}^{2} (y + 2)}$

Add $2 y$ and $4 y$ .

$\frac{2 (y^{2} + 6 y - 4)}{{(y - 1)}^{2} (y + 2)}$

Add (2y)/(y^2-2y+1)+8/(y^2+1y-2)

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