Add b/(b^2+4b+4)+9/(b^2+7b+10)

$\frac{b}{b^{2} + 4 b + 4} + \frac{9}{b^{2} + 7 b + 10}$

Simplify each term.

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Factor using the perfect square rule.

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Rewrite $4$ as $2^{2}$ .

$\frac{b}{b^{2} + 4 b + 2^{2}} + \frac{9}{b^{2} + 7 b + 10}$

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

$2 a b = 2 \cdot b \cdot 2$

Simplify.

$2 a b = 4 b$

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

$\frac{b}{{(b + 2)}^{2}} + \frac{9}{b^{2} + 7 b + 10}$

Factor $b^{2} + 7 b + 10$ 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 $10$ and whose sum is $7$ .

$2, 5$

Write the factored form using these integers.

$\frac{b}{{(b + 2)}^{2}} + \frac{9}{(b + 2) (b + 5)}$

To write $\frac{b}{{(b + 2)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{b + 5}{b + 5}$ .

$\frac{b}{{(b + 2)}^{2}} \cdot \frac{b + 5}{b + 5} + \frac{9}{(b + 2) (b + 5)}$

To write $\frac{9}{(b + 2) (b + 5)}$ as a fraction with a common denominator, multiply by $\frac{b + 2}{b + 2}$ .

$\frac{b}{{(b + 2)}^{2}} \cdot \frac{b + 5}{b + 5} + \frac{9}{(b + 2) (b + 5)} \cdot \frac{b + 2}{b + 2}$

Write each expression with a common denominator of $(b + 5) {(b + 2)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{b}{{(b + 2)}^{2}}$ and $\frac{b + 5}{b + 5}$ .

$\frac{b (b + 5)}{{(b + 2)}^{2} (b + 5)} + \frac{9}{(b + 2) (b + 5)} \cdot \frac{b + 2}{b + 2}$

Multiply $\frac{9}{(b + 2) (b + 5)}$ and $\frac{b + 2}{b + 2}$ .

$\frac{b (b + 5)}{{(b + 2)}^{2} (b + 5)} + \frac{9 (b + 2)}{(b + 2) (b + 5) (b + 2)}$

Raise $b + 2$ to the power of $1$ .

$\frac{b (b + 5)}{{(b + 2)}^{2} (b + 5)} + \frac{9 (b + 2)}{{(b + 2)}^{1} (b + 2) (b + 5)}$

Raise $b + 2$ to the power of $1$ .

$\frac{b (b + 5)}{{(b + 2)}^{2} (b + 5)} + \frac{9 (b + 2)}{{(b + 2)}^{1} {(b + 2)}^{1} (b + 5)}$

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

$\frac{b (b + 5)}{{(b + 2)}^{2} (b + 5)} + \frac{9 (b + 2)}{{(b + 2)}^{1 + 1} (b + 5)}$

Add $1$ and $1$ .

$\frac{b (b + 5)}{{(b + 2)}^{2} (b + 5)} + \frac{9 (b + 2)}{{(b + 2)}^{2} (b + 5)}$

Combine the numerators over the common denominator.

$\frac{b (b + 5) + 9 (b + 2)}{{(b + 2)}^{2} (b + 5)}$

Simplify the numerator.

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Apply the distributive property.

$\frac{b \cdot b + b \cdot 5 + 9 (b + 2)}{{(b + 2)}^{2} (b + 5)}$

Multiply $b$ by $b$ .

$\frac{b^{2} + b \cdot 5 + 9 (b + 2)}{{(b + 2)}^{2} (b + 5)}$

Move $5$ to the left of $b$ .

$\frac{b^{2} + 5 \cdot b + 9 (b + 2)}{{(b + 2)}^{2} (b + 5)}$

Apply the distributive property.

$\frac{b^{2} + 5 b + 9 b + 9 \cdot 2}{{(b + 2)}^{2} (b + 5)}$

Multiply $9$ by $2$ .

$\frac{b^{2} + 5 b + 9 b + 18}{{(b + 2)}^{2} (b + 5)}$

Add $5 b$ and $9 b$ .

$\frac{b^{2} + 14 b + 18}{{(b + 2)}^{2} (b + 5)}$

Add b/(b^2+4b+4)+9/(b^2+7b+10)