Divide ((5x^2+10x)/(x^2+2x+1))÷((20x+40)/(x^2-1))

$\frac{5 x^{2} + 10 x}{x^{2} + 2 x + 1} \div \frac{20 x + 40}{x^{2} - 1}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{5 x^{2} + 10 x}{x^{2} + 2 x + 1} \cdot \frac{x^{2} - 1}{20 x + 40}$

Step 2

Factor $5 x$ out of $5 x^{2} + 10 x$ .

Factor $5 x$ out of $5 x^{2}$ .

$\frac{5 x (x) + 10 x}{x^{2} + 2 x + 1} \cdot \frac{x^{2} - 1}{20 x + 40}$

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

$\frac{5 x (x) + 5 x (2)}{x^{2} + 2 x + 1} \cdot \frac{x^{2} - 1}{20 x + 40}$

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

$\frac{5 x (x + 2)}{x^{2} + 2 x + 1} \cdot \frac{x^{2} - 1}{20 x + 40}$

Step 3

Factor using the perfect square rule.

Rewrite $1$ as $1^{2}$ .

$\frac{5 x (x + 2)}{x^{2} + 2 x + 1^{2}} \cdot \frac{x^{2} - 1}{20 x + 40}$

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

$2 x = 2 \cdot x \cdot 1$

Rewrite the polynomial.

$\frac{5 x (x + 2)}{x^{2} + 2 \cdot x \cdot 1 + 1^{2}} \cdot \frac{x^{2} - 1}{20 x + 40}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 1$ .

$\frac{5 x (x + 2)}{{(x + 1)}^{2}} \cdot \frac{x^{2} - 1}{20 x + 40}$

Step 4

Simplify the numerator.

Rewrite $1$ as $1^{2}$ .

$\frac{5 x (x + 2)}{{(x + 1)}^{2}} \cdot \frac{x^{2} - 1^{2}}{20 x + 40}$

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 $b = 1$ .

$\frac{5 x (x + 2)}{{(x + 1)}^{2}} \cdot \frac{(x + 1) (x - 1)}{20 x + 40}$

Step 5

Factor $20$ out of $20 x + 40$ .

Factor $20$ out of $20 x$ .

$\frac{5 x (x + 2)}{{(x + 1)}^{2}} \cdot \frac{(x + 1) (x - 1)}{20 (x) + 40}$

Factor $20$ out of $40$ .

$\frac{5 x (x + 2)}{{(x + 1)}^{2}} \cdot \frac{(x + 1) (x - 1)}{20 x + 20 \cdot 2}$

Factor $20$ out of $20 x + 20 \cdot 2$ .

$\frac{5 x (x + 2)}{{(x + 1)}^{2}} \cdot \frac{(x + 1) (x - 1)}{20 (x + 2)}$

Step 6

Combine.

$\frac{5 x (x + 2) ((x + 1) (x - 1))}{{(x + 1)}^{2} (20 (x + 2))}$

Step 7

Cancel the common factor of $5$ and $20$ .

Factor $5$ out of $5 x (x + 2) ((x + 1) (x - 1))$ .

$\frac{5 ((x (x + 2)) ((x + 1) (x - 1)))}{{(x + 1)}^{2} (20 (x + 2))}$

Cancel the common factors.

Factor $5$ out of ${(x + 1)}^{2} (20 (x + 2))$ .

$\frac{5 ((x (x + 2)) ((x + 1) (x - 1)))}{5 ({(x + 1)}^{2} (4 (x + 2)))}$

Cancel the common factor.

$\frac{5 ((x (x + 2)) ((x + 1) (x - 1)))}{5 ({(x + 1)}^{2} (4 (x + 2)))}$

Rewrite the expression.

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

Step 8

Cancel the common factor of $x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 9

Cancel the common factor of $x + 1$ and ${(x + 1)}^{2}$ .

Factor $x + 1$ out of $(x) ((x + 1) (x - 1))$ .

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

Cancel the common factors.

Factor $x + 1$ out of ${(x + 1)}^{2} \cdot (4)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 10

Move $4$ to the left of $x + 1$ .

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

Divide ((5x^2+10x)/(x^2+2x+1))÷((20x+40)/(x^2-1))

Related Questions

Support LCMGCF.com by disabling your adblocker.