Find the Product (x^2-16)/(2x+8)*(x^3-2x^2+x)/(x^2+3x-4)

$\frac{x^{2} - 16}{2 x + 8} \cdot \frac{x^{3} - 2 x^{2} + x}{x^{2} + 3 x - 4}$

Simplify the numerator.

Rewrite $16$ as $4^{2}$ .

$\frac{x^{2} - 4^{2}}{2 x + 8} \cdot \frac{x^{3} - 2 x^{2} + x}{x^{2} + 3 x - 4}$

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 $b = 4$ .

$\frac{(x + 4) (x - 4)}{2 x + 8} \cdot \frac{x^{3} - 2 x^{2} + x}{x^{2} + 3 x - 4}$

Factor $2$ out of $2 x + 8$ .

Factor $2$ out of $2 x$ .

$\frac{(x + 4) (x - 4)}{2 (x) + 8} \cdot \frac{x^{3} - 2 x^{2} + x}{x^{2} + 3 x - 4}$

Factor $2$ out of $8$ .

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

Factor $2$ out of $2 x + 2 \cdot 4$ .

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

Simplify the numerator.

Factor $x$ out of $x^{3} - 2 x^{2} + x$ .

Factor $x$ out of $x^{3}$ .

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

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

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

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

$\frac{(x + 4) (x - 4)}{2 (x + 4)} \cdot \frac{x \cdot x^{2} + x (- 2 x) + x^{1}}{x^{2} + 3 x - 4}$

Factor $x$ out of $x^{1}$ .

$\frac{(x + 4) (x - 4)}{2 (x + 4)} \cdot \frac{x \cdot x^{2} + x (- 2 x) + x \cdot 1}{x^{2} + 3 x - 4}$

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

$\frac{(x + 4) (x - 4)}{2 (x + 4)} \cdot \frac{x (x^{2} - 2 x) + x \cdot 1}{x^{2} + 3 x - 4}$

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

$\frac{(x + 4) (x - 4)}{2 (x + 4)} \cdot \frac{x (x^{2} - 2 x + 1)}{x^{2} + 3 x - 4}$

Factor using the perfect square rule.

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

$\frac{(x + 4) (x - 4)}{2 (x + 4)} \cdot \frac{x (x^{2} - 2 x + 1^{2})}{x^{2} + 3 x - 4}$

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.

$\frac{(x + 4) (x - 4)}{2 (x + 4)} \cdot \frac{x (x^{2} - 2 \cdot x \cdot 1 + 1^{2})}{x^{2} + 3 x - 4}$

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

$\frac{(x + 4) (x - 4)}{2 (x + 4)} \cdot \frac{x {(x - 1)}^{2}}{x^{2} + 3 x - 4}$

Factor $x^{2} + 3 x - 4$ 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 $- 4$ and whose sum is $3$ .

$- 1, 4$

Write the factored form using these integers.

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

Simplify terms.

Combine.

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

Cancel the common factor of $x + 4$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

Factor $x - 1$ out of $(x - 4) (x {(x - 1)}^{2})$ .

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

Cancel the common factors.

Factor $x - 1$ out of $2 ((x - 1) (x + 4))$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Reorder factors in $\frac{(x - 4) x (x - 1)}{2 (x + 4)}$ .

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

Find the Product (x^2-16)/(2x+8)*(x^3-2x^2+x)/(x^2+3x-4)

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