Divide ((x^2+10x+25)/(x-5))÷((x^2-25)/(5x+10))

$\frac{x^{2} + 10 x + 25}{x - 5} \div \frac{x^{2} - 25}{5 x + 10}$

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + 10 x + 25}{x - 5} \cdot \frac{5 x + 10}{x^{2} - 25}$

Factor using the perfect square rule.

Rewrite $25$ as $5^{2}$ .

$\frac{x^{2} + 10 x + 5^{2}}{x - 5} \cdot \frac{5 x + 10}{x^{2} - 25}$

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

$10 x = 2 \cdot x \cdot 5$

Rewrite the polynomial.

$\frac{x^{2} + 2 \cdot x \cdot 5 + 5^{2}}{x - 5} \cdot \frac{5 x + 10}{x^{2} - 25}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 5$ .

$\frac{{(x + 5)}^{2}}{x - 5} \cdot \frac{5 x + 10}{x^{2} - 25}$

Factor $5$ out of $5 x + 10$ .

Factor $5$ out of $5 x$ .

$\frac{{(x + 5)}^{2}}{x - 5} \cdot \frac{5 (x) + 10}{x^{2} - 25}$

Factor $5$ out of $10$ .

$\frac{{(x + 5)}^{2}}{x - 5} \cdot \frac{5 x + 5 \cdot 2}{x^{2} - 25}$

Factor $5$ out of $5 x + 5 \cdot 2$ .

$\frac{{(x + 5)}^{2}}{x - 5} \cdot \frac{5 (x + 2)}{x^{2} - 25}$

Simplify the denominator.

Rewrite $25$ as $5^{2}$ .

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

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 $b = 5$ .

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

Cancel the common factor of $x + 5$ .

Factor $x + 5$ out of ${(x + 5)}^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{x + 5}{x - 5}$ by $\frac{5 (x + 2)}{x - 5}$ .

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

Raise $x - 5$ to the power of $1$ .

$\frac{(x + 5) (5 (x + 2))}{{(x - 5)}^{1} (x - 5)}$

Raise $x - 5$ to the power of $1$ .

$\frac{(x + 5) (5 (x + 2))}{{(x - 5)}^{1} {(x - 5)}^{1}}$

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

$\frac{(x + 5) (5 (x + 2))}{{(x - 5)}^{1 + 1}}$

Add $1$ and $1$ .

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

Move $5$ to the left of $x + 5$ .

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

Apply the distributive property.

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

Multiply $5$ by $5$ .

$\frac{(5 x + 25) (x + 2)}{{(x - 5)}^{2}}$

Expand $(5 x + 25) (x + 2)$ using the FOIL Method.

Apply the distributive property.

$\frac{5 x (x + 2) + 25 (x + 2)}{{(x - 5)}^{2}}$

Apply the distributive property.

$\frac{5 x \cdot x + 5 x \cdot 2 + 25 (x + 2)}{{(x - 5)}^{2}}$

Apply the distributive property.

$\frac{5 x \cdot x + 5 x \cdot 2 + 25 x + 25 \cdot 2}{{(x - 5)}^{2}}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{5 (x \cdot x) + 5 x \cdot 2 + 25 x + 25 \cdot 2}{{(x - 5)}^{2}}$

Multiply $x$ by $x$ .

$\frac{5 x^{2} + 5 x \cdot 2 + 25 x + 25 \cdot 2}{{(x - 5)}^{2}}$

Multiply $2$ by $5$ .

$\frac{5 x^{2} + 10 x + 25 x + 25 \cdot 2}{{(x - 5)}^{2}}$

Multiply $25$ by $2$ .

$\frac{5 x^{2} + 10 x + 25 x + 50}{{(x - 5)}^{2}}$

Add $10 x$ and $25 x$ .

$\frac{5 x^{2} + 35 x + 50}{{(x - 5)}^{2}}$

Split the fraction $\frac{5 x^{2} + 35 x + 50}{{(x - 5)}^{2}}$ into two fractions.

$\frac{5 x^{2} + 35 x}{{(x - 5)}^{2}} + \frac{50}{{(x - 5)}^{2}}$

Split the fraction $\frac{5 x^{2} + 35 x}{{(x - 5)}^{2}}$ into two fractions.

$\frac{5 x^{2}}{{(x - 5)}^{2}} + \frac{35 x}{{(x - 5)}^{2}} + \frac{50}{{(x - 5)}^{2}}$

Divide ((x^2+10x+25)/(x-5))÷((x^2-25)/(5x+10))

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