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

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Factor $2 x$ out of $2 x^{2} + 4 x$ .

Factor $2 x$ out of $2 x^{2}$ .

$\frac{2 x (x) + 4 x}{x^{2} - 4} \cdot \frac{3 x - 6}{x^{2} - 3 x + 2}$

Factor $2 x$ out of $4 x$ .

$\frac{2 x (x) + 2 x (2)}{x^{2} - 4} \cdot \frac{3 x - 6}{x^{2} - 3 x + 2}$

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

$\frac{2 x (x + 2)}{x^{2} - 4} \cdot \frac{3 x - 6}{x^{2} - 3 x + 2}$

Step 3

Simplify the denominator.

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

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

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 $b = 2$ .

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

Step 4

Simplify terms.

Cancel the common factor of $x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Factor $3$ out of $3 x - 6$ .

Factor $3$ out of $3 x$ .

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

Factor $3$ out of $- 6$ .

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

Factor $3$ out of $3 x + 3 \cdot - 2$ .

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

Step 5

Factor $x^{2} - 3 x + 2$ 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 $2$ and whose sum is $- 3$ .

$- 2, - 1$

Write the factored form using these integers.

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

Step 6

Simplify terms.

Cancel the common factor of $x - 2$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine $2$ and $\frac{3}{(x - 2) (x - 1)}$ .

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

Multiply $2$ by $3$ .

$x \frac{6}{(x - 2) (x - 1)}$

Combine $x$ and $\frac{6}{(x - 2) (x - 1)}$ .

$\frac{x \cdot 6}{(x - 2) (x - 1)}$

Move $6$ to the left of $x$ .

$\frac{6 x}{(x - 2) (x - 1)}$

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

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