Simplify 5/(x+3)+(2x)/(x^2+6x+9)

$\frac{5}{x + 3} + \frac{2 x}{x^{2} + 6 x + 9}$

Step 1

Factor using the perfect square rule.

Rewrite $9$ as $3^{2}$ .

$\frac{5}{x + 3} + \frac{2 x}{x^{2} + 6 x + 3^{2}}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

Rewrite the polynomial.

$\frac{5}{x + 3} + \frac{2 x}{x^{2} + 2 \cdot x \cdot 3 + 3^{2}}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 3$ .

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

Step 2

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

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

Step 3

Write each expression with a common denominator of ${(x + 3)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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

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

Raise $x + 3$ to the power of $1$ .

$\frac{5 (x + 3)}{{(x + 3)}^{1} (x + 3)} + \frac{2 x}{{(x + 3)}^{2}}$

Raise $x + 3$ to the power of $1$ .

$\frac{5 (x + 3)}{{(x + 3)}^{1} {(x + 3)}^{1}} + \frac{2 x}{{(x + 3)}^{2}}$

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

$\frac{5 (x + 3)}{{(x + 3)}^{1 + 1}} + \frac{2 x}{{(x + 3)}^{2}}$

Add $1$ and $1$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Apply the distributive property.

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

Multiply $5$ by $3$ .

$\frac{5 x + 15 + 2 x}{{(x + 3)}^{2}}$

Add $5 x$ and $2 x$ .

$\frac{7 x + 15}{{(x + 3)}^{2}}$