Simplify (3y)/(y^2+7y+10)+2/(y+2)

$\frac{3 y}{y^{2} + 7 y + 10} + \frac{2}{y + 2}$

Step 1

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

$2, 5$

Write the factored form using these integers.

$\frac{3 y}{(y + 2) (y + 5)} + \frac{2}{y + 2}$

Step 2

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

$\frac{3 y}{(y + 2) (y + 5)} + \frac{2}{y + 2} \cdot \frac{y + 5}{y + 5}$

Step 3

Simplify terms.

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

$\frac{3 y}{(y + 2) (y + 5)} + \frac{2 (y + 5)}{(y + 2) (y + 5)}$

Combine the numerators over the common denominator.

$\frac{3 y + 2 (y + 5)}{(y + 2) (y + 5)}$

Step 4

Simplify the numerator.

Apply the distributive property.

$\frac{3 y + 2 y + 2 \cdot 5}{(y + 2) (y + 5)}$

Multiply $2$ by $5$ .

$\frac{3 y + 2 y + 10}{(y + 2) (y + 5)}$

Add $3 y$ and $2 y$ .

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

Factor $5$ out of $5 y + 10$ .

Factor $5$ out of $5 y$ .

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

Factor $5$ out of $10$ .

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

Factor $5$ out of $5 y + 5 \cdot 2$ .

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

Step 5

Cancel the common factor of $y + 2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{5}{y + 5}$

Simplify (3y)/(y^2+7y+10)+2/(y+2)

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