Simplify (9x^2-9)/(x^4+3x^3-4x^2)*(x+4)/(x+1)

$\frac{9 x^{2} - 9}{x^{4} + 3 x^{3} - 4 x^{2}} \cdot \frac{x + 4}{x + 1}$

Step 1

Simplify the numerator.

Factor $9$ out of $9 x^{2} - 9$ .

Factor $9$ out of $9 x^{2}$ .

$\frac{9 (x^{2}) - 9}{x^{4} + 3 x^{3} - 4 x^{2}} \cdot \frac{x + 4}{x + 1}$

Factor $9$ out of $- 9$ .

$\frac{9 (x^{2}) + 9 (- 1)}{x^{4} + 3 x^{3} - 4 x^{2}} \cdot \frac{x + 4}{x + 1}$

Factor $9$ out of $9 (x^{2}) + 9 (- 1)$ .

$\frac{9 (x^{2} - 1)}{x^{4} + 3 x^{3} - 4 x^{2}} \cdot \frac{x + 4}{x + 1}$

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

$\frac{9 (x^{2} - 1^{2})}{x^{4} + 3 x^{3} - 4 x^{2}} \cdot \frac{x + 4}{x + 1}$

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 $b = 1$ .

$\frac{9 (x + 1) (x - 1)}{x^{4} + 3 x^{3} - 4 x^{2}} \cdot \frac{x + 4}{x + 1}$

Step 2

Simplify the denominator.

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

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

$\frac{9 (x + 1) (x - 1)}{x^{2} x^{2} + 3 x^{3} - 4 x^{2}} \cdot \frac{x + 4}{x + 1}$

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

$\frac{9 (x + 1) (x - 1)}{x^{2} x^{2} + x^{2} (3 x) - 4 x^{2}} \cdot \frac{x + 4}{x + 1}$

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

$\frac{9 (x + 1) (x - 1)}{x^{2} x^{2} + x^{2} (3 x) + x^{2} \cdot - 4} \cdot \frac{x + 4}{x + 1}$

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

$\frac{9 (x + 1) (x - 1)}{x^{2} (x^{2} + 3 x) + x^{2} \cdot - 4} \cdot \frac{x + 4}{x + 1}$

Factor $x^{2}$ out of $x^{2} (x^{2} + 3 x) + x^{2} \cdot - 4$ .

$\frac{9 (x + 1) (x - 1)}{x^{2} (x^{2} + 3 x - 4)} \cdot \frac{x + 4}{x + 1}$

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.

$\frac{9 (x + 1) (x - 1)}{x^{2} ((x - 1) (x + 4))} \cdot \frac{x + 4}{x + 1}$

$\frac{9 (x + 1) (x - 1)}{x^{2} (x - 1) (x + 4)} \cdot \frac{x + 4}{x + 1}$

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x + 1$ .

Factor $x + 1$ out of $9 (x + 1) (x - 1)$ .

$\frac{(x + 1) (9 (x - 1))}{x^{2} (x - 1) (x + 4)} \cdot \frac{x + 4}{x + 1}$

Cancel the common factor.

$\frac{(x + 1) (9 (x - 1))}{x^{2} (x - 1) (x + 4)} \cdot \frac{x + 4}{x + 1}$

Rewrite the expression.

$\frac{9 (x - 1)}{x^{2} (x - 1) (x + 4)} \cdot (x + 4)$

Cancel the common factor of $x + 4$ .

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

$\frac{9 (x - 1)}{(x + 4) (x^{2} (x - 1))} \cdot (x + 4)$

Cancel the common factor.

$\frac{9 (x - 1)}{(x + 4) (x^{2} (x - 1))} \cdot (x + 4)$

Rewrite the expression.

$\frac{9 (x - 1)}{x^{2} (x - 1)}$

Cancel the common factor of $x - 1$ .

Cancel the common factor.

$\frac{9 (x - 1)}{x^{2} (x - 1)}$

Rewrite the expression.

$\frac{9}{x^{2}}$

Simplify (9x^2-9)/(x^4+3x^3-4x^2)*(x+4)/(x+1)

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