Simplify (x+1)/(x-6)*(x^2-16)/(x^3+5x^2+4x)

$\frac{x + 1}{x - 6} \cdot \frac{x^{2} - 16}{x^{3} + 5 x^{2} + 4 x}$

Step 1

Simplify the numerator.

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

$\frac{x + 1}{x - 6} \cdot \frac{x^{2} - 4^{2}}{x^{3} + 5 x^{2} + 4 x}$

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 $b = 4$ .

$\frac{x + 1}{x - 6} \cdot \frac{(x + 4) (x - 4)}{x^{3} + 5 x^{2} + 4 x}$

Step 2

Simplify the denominator.

Factor $x$ out of $x^{3} + 5 x^{2} + 4 x$ .

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

$\frac{x + 1}{x - 6} \cdot \frac{(x + 4) (x - 4)}{x \cdot x^{2} + 5 x^{2} + 4 x}$

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

$\frac{x + 1}{x - 6} \cdot \frac{(x + 4) (x - 4)}{x \cdot x^{2} + x (5 x) + 4 x}$

Factor $x$ out of $4 x$ .

$\frac{x + 1}{x - 6} \cdot \frac{(x + 4) (x - 4)}{x \cdot x^{2} + x (5 x) + x \cdot 4}$

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

$\frac{x + 1}{x - 6} \cdot \frac{(x + 4) (x - 4)}{x (x^{2} + 5 x) + x \cdot 4}$

Factor $x$ out of $x (x^{2} + 5 x) + x \cdot 4$ .

$\frac{x + 1}{x - 6} \cdot \frac{(x + 4) (x - 4)}{x (x^{2} + 5 x + 4)}$

Factor $x^{2} + 5 x + 4$ 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 $5$ .

$1, 4$

Write the factored form using these integers.

$\frac{x + 1}{x - 6} \cdot \frac{(x + 4) (x - 4)}{x ((x + 1) (x + 4))}$

$\frac{x + 1}{x - 6} \cdot \frac{(x + 4) (x - 4)}{x (x + 1) (x + 4)}$

Step 3

Simplify terms.

Cancel the common factor of $x + 1$ .

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

$\frac{x + 1}{x - 6} \cdot \frac{(x + 4) (x - 4)}{(x + 1) (x (x + 4))}$

Cancel the common factor.

$\frac{x + 1}{x - 6} \cdot \frac{(x + 4) (x - 4)}{(x + 1) (x (x + 4))}$

Rewrite the expression.

$\frac{1}{x - 6} \cdot \frac{(x + 4) (x - 4)}{x (x + 4)}$

Multiply $\frac{1}{x - 6}$ by $\frac{(x + 4) (x - 4)}{x (x + 4)}$ .

$\frac{(x + 4) (x - 4)}{(x - 6) (x (x + 4))}$

Cancel the common factor of $x + 4$ .

Cancel the common factor.

$\frac{(x + 4) (x - 4)}{(x - 6) (x (x + 4))}$

Rewrite the expression.

$\frac{x - 4}{(x - 6) (x)}$

Reorder factors in $\frac{x - 4}{(x - 6) (x)}$ .

$\frac{x - 4}{x (x - 6)}$

Simplify (x+1)/(x-6)*(x^2-16)/(x^3+5x^2+4x)

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