Find the Product (4x^2-8x)/(x+1)*(x^2+2x+1)/(4x)

$\frac{4 x^{2} - 8 x}{x + 1} \cdot \frac{x^{2} + 2 x + 1}{4 x}$

Step 1

Factor $4 x$ out of $4 x^{2} - 8 x$ .

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

$\frac{4 x (x) - 8 x}{x + 1} \cdot \frac{x^{2} + 2 x + 1}{4 x}$

Factor $4 x$ out of $- 8 x$ .

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

Factor $4 x$ out of $4 x (x) + 4 x (- 2)$ .

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

Step 2

Factor using the perfect square rule.

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

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

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.

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

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

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

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of $4 x$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 1$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

$(x - 2) \cdot (x + 1)$

Step 4

Expand $(x - 2) (x + 1)$ using the FOIL Method.

Apply the distributive property.

$x (x + 1) - 2 (x + 1)$

Apply the distributive property.

$x \cdot x + x \cdot 1 - 2 (x + 1)$

Apply the distributive property.

$x \cdot x + x \cdot 1 - 2 x - 2 \cdot 1$

Step 5

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot 1 - 2 x - 2 \cdot 1$

Multiply $x$ by $1$ .

$x^{2} + x - 2 x - 2 \cdot 1$

Multiply $- 2$ by $1$ .

$x^{2} + x - 2 x - 2$

Subtract $2 x$ from $x$ .

$x^{2} - x - 2$

Find the Product (4x^2-8x)/(x+1)*(x^2+2x+1)/(4x)

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