Simplify 5/(y+10)+(4y)/(y^2+12y+20)

$\frac{5}{y + 10} + \frac{4 y}{y^{2} + 12 y + 20}$

Step 1

Factor $y^{2} + 12 y + 20$ using the AC method.

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 $20$ and whose sum is $12$ .

$2, 10$

Write the factored form using these integers.

$\frac{5}{y + 10} + \frac{4 y}{(y + 2) (y + 10)}$

Step 2

To write $\frac{5}{y + 10}$ as a fraction with a common denominator, multiply by $\frac{y + 2}{y + 2}$ .

$\frac{5}{y + 10} \cdot \frac{y + 2}{y + 2} + \frac{4 y}{(y + 2) (y + 10)}$

Step 3

Write each expression with a common denominator of $(y + 10) (y + 2)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5}{y + 10}$ by $\frac{y + 2}{y + 2}$ .

$\frac{5 (y + 2)}{(y + 10) (y + 2)} + \frac{4 y}{(y + 2) (y + 10)}$

Reorder the factors of $(y + 2) (y + 10)$ .

$\frac{5 (y + 2)}{(y + 10) (y + 2)} + \frac{4 y}{(y + 10) (y + 2)}$

Step 4

Combine the numerators over the common denominator.

$\frac{5 (y + 2) + 4 y}{(y + 10) (y + 2)}$

Step 5

Simplify the numerator.

Apply the distributive property.

$\frac{5 y + 5 \cdot 2 + 4 y}{(y + 10) (y + 2)}$

Multiply $5$ by $2$ .

$\frac{5 y + 10 + 4 y}{(y + 10) (y + 2)}$

Add $5 y$ and $4 y$ .

$\frac{9 y + 10}{(y + 10) (y + 2)}$

Simplify 5/(y+10)+(4y)/(y^2+12y+20)

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