Simplify (6x)/(x^2+4x+4)+3/(x+2)

$\frac{6 x}{x^{2} + 4 x + 4} + \frac{3}{x + 2}$

Step 1

Factor using the perfect square rule.

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

$\frac{6 x}{x^{2} + 4 x + 2^{2}} + \frac{3}{x + 2}$

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

$4 x = 2 \cdot x \cdot 2$

Rewrite the polynomial.

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

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

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

Step 2

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

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

Step 3

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

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

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

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

$\frac{6 x}{{(x + 2)}^{2}} + \frac{3 (x + 2)}{{(x + 2)}^{1} (x + 2)}$

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

$\frac{6 x}{{(x + 2)}^{2}} + \frac{3 (x + 2)}{{(x + 2)}^{1} {(x + 2)}^{1}}$

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

$\frac{6 x}{{(x + 2)}^{2}} + \frac{3 (x + 2)}{{(x + 2)}^{1 + 1}}$

Add $1$ and $1$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Factor $3$ out of $6 x + 3 (x + 2)$ .

Factor $3$ out of $6 x$ .

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

Factor $3$ out of $3 (2 x) + 3 (x + 2)$ .

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

Add $2 x$ and $x$ .

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