Simplify 3/(x+5)+(2x)/(x^2+7x+10)

$\frac{3}{x + 5} + \frac{2 x}{x^{2} + 7 x + 10}$

Step 1

Factor $x^{2} + 7 x + 10$ 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.

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

Step 2

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

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

Step 3

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

Multiply $\frac{3}{x + 5}$ by $\frac{x + 2}{x + 2}$ .

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

Reorder the factors of $(x + 5) (x + 2)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Apply the distributive property.

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

Multiply $3$ by $2$ .

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

Add $3 x$ and $2 x$ .

$\frac{5 x + 6}{(x + 2) (x + 5)}$

Simplify 3/(x+5)+(2x)/(x^2+7x+10)

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