Simplify (x^2-4y^2)÷((5x-10y)/x)

$(x^{2} - 4 y^{2}) \div \frac{5 x - 10 y}{x}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$(x^{2} - 4 y^{2}) \frac{x}{5 x - 10 y}$

Step 2

Simplify terms.

Factor $5$ out of $5 x - 10 y$ .

Factor $5$ out of $5 x$ .

$(x^{2} - 4 y^{2}) \frac{x}{5 (x) - 10 y}$

Factor $5$ out of $- 10 y$ .

$(x^{2} - 4 y^{2}) \frac{x}{5 (x) + 5 (- 2 y)}$

Factor $5$ out of $5 (x) + 5 (- 2 y)$ .

$(x^{2} - 4 y^{2}) \frac{x}{5 (x - 2 y)}$

Multiply $x^{2} - 4 y^{2}$ by $\frac{x}{5 (x - 2 y)}$ .

$\frac{(x^{2} - 4 y^{2}) x}{5 (x - 2 y)}$

Step 3

Simplify the numerator.

Rewrite $4 y^{2}$ as ${(2 y)}^{2}$ .

$\frac{(x^{2} - {(2 y)}^{2}) x}{5 (x - 2 y)}$

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 $b = 2 y$ .

$\frac{(x + 2 y) (x - (2 y)) x}{5 (x - 2 y)}$

Multiply $2$ by $- 1$ .

$\frac{(x + 2 y) (x - 2 y) x}{5 (x - 2 y)}$

Step 4

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x - 2 y$ .

Cancel the common factor.

$\frac{(x + 2 y) (x - 2 y) x}{5 (x - 2 y)}$

Rewrite the expression.

$\frac{(x + 2 y) x}{5}$

Reorder factors in $\frac{(x + 2 y) x}{5}$ .

$\frac{x (x + 2 y)}{5}$