Simplify ((x^2-4)/(x+3))÷((x^2-4x+4)/(4x+12))

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Simplify the numerator.

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

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

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 $b = 2$ .

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

Step 3

Factor $4$ out of $4 x + 12$ .

Factor $4$ out of $4 x$ .

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

Factor $4$ out of $12$ .

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

Factor $4$ out of $4 x + 4 \cdot 3$ .

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

Step 4

Factor using the perfect square rule.

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

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

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.

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

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

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

Step 5

Simplify terms.

Cancel the common factor of $x - 2$ .

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

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

Factor $x - 2$ out of ${(x - 2)}^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 3$ .

Factor $x + 3$ out of $4 (x + 3)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $x + 2$ by $\frac{4}{x - 2}$ .

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

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

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

Simplify ((x^2-4)/(x+3))÷((x^2-4x+4)/(4x+12))

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