Divide ((3x^2+13x+4)/(x^2-4))÷((4x+16)/(x+2))

$\frac{3 x^{2} + 13 x + 4}{x^{2} - 4} \div \frac{4 x + 16}{x + 2}$

To divide by a fraction, multiply by its reciprocal.

$\frac{3 x^{2} + 13 x + 4}{x^{2} - 4} \cdot \frac{x + 2}{4 x + 16}$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 4 = 12$ and whose sum is $b = 13$ .

Factor $13$ out of $13 x$ .

$\frac{3 x^{2} + 13 (x) + 4}{x^{2} - 4} \cdot \frac{x + 2}{4 x + 16}$

Rewrite $13$ as $1$ plus $12$

$\frac{3 x^{2} + (1 + 12) x + 4}{x^{2} - 4} \cdot \frac{x + 2}{4 x + 16}$

Apply the distributive property.

$\frac{3 x^{2} + 1 x + 12 x + 4}{x^{2} - 4} \cdot \frac{x + 2}{4 x + 16}$

Multiply $x$ by $1$ .

$\frac{3 x^{2} + x + 12 x + 4}{x^{2} - 4} \cdot \frac{x + 2}{4 x + 16}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(3 x^{2} + x) + 12 x + 4}{x^{2} - 4} \cdot \frac{x + 2}{4 x + 16}$

Factor out the greatest common factor (GCF) from each group.

$\frac{x (3 x + 1) + 4 (3 x + 1)}{x^{2} - 4} \cdot \frac{x + 2}{4 x + 16}$

Factor the polynomial by factoring out the greatest common factor, $3 x + 1$ .

$\frac{(3 x + 1) (x + 4)}{x^{2} - 4} \cdot \frac{x + 2}{4 x + 16}$

Simplify the denominator.

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

$\frac{(3 x + 1) (x + 4)}{x^{2} - 2^{2}} \cdot \frac{x + 2}{4 x + 16}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$\frac{(3 x + 1) (x + 4)}{(x + 2) (x - 2)} \cdot \frac{x + 2}{4 x + 16}$

Cancel the common factor of $x + 2$ .

Cancel the common factor.

$\frac{(3 x + 1) (x + 4)}{(x + 2) (x - 2)} \cdot \frac{x + 2}{4 x + 16}$

Rewrite the expression.

$\frac{(3 x + 1) (x + 4)}{x - 2} \cdot \frac{1}{4 x + 16}$

Multiply $\frac{(3 x + 1) (x + 4)}{x - 2}$ by $\frac{1}{4 x + 16}$ .

$\frac{(3 x + 1) (x + 4)}{(x - 2) (4 x + 16)}$

Factor $4$ out of $4 x + 16$ .

Factor $4$ out of $4 x$ .

$\frac{(3 x + 1) (x + 4)}{(x - 2) (4 (x) + 16)}$

Factor $4$ out of $16$ .

$\frac{(3 x + 1) (x + 4)}{(x - 2) (4 x + 4 \cdot 4)}$

Factor $4$ out of $4 x + 4 \cdot 4$ .

$\frac{(3 x + 1) (x + 4)}{(x - 2) (4 (x + 4))}$

$\frac{(3 x + 1) (x + 4)}{(x - 2) \cdot 4 (x + 4)}$

Cancel the common factor of $x + 4$ .

Cancel the common factor.

$\frac{(3 x + 1) (x + 4)}{(x - 2) \cdot 4 (x + 4)}$

Rewrite the expression.

$\frac{3 x + 1}{(x - 2) \cdot 4}$

Move $4$ to the left of $x - 2$ .

$\frac{3 x + 1}{4 (x - 2)}$

Divide ((3x^2+13x+4)/(x^2-4))÷((4x+16)/(x+2))

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