Simplify ((x^2+9x+8)/(x^2))÷((x^2-1)/(x^3+3x^2-4x))

$\frac{x^{2} + 9 x + 8}{x^{2}} \div \frac{x^{2} - 1}{x^{3} + 3 x^{2} - 4 x}$

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + 9 x + 8}{x^{2}} \cdot \frac{x^{3} + 3 x^{2} - 4 x}{x^{2} - 1}$

Factor $x^{2} + 9 x + 8$ 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 $8$ and whose sum is $9$ .

$1, 8$

Write the factored form using these integers.

$\frac{(x + 1) (x + 8)}{x^{2}} \cdot \frac{x^{3} + 3 x^{2} - 4 x}{x^{2} - 1}$

Simplify the numerator.

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

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

$\frac{(x + 1) (x + 8)}{x^{2}} \cdot \frac{x \cdot x^{2} + 3 x^{2} - 4 x}{x^{2} - 1}$

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

$\frac{(x + 1) (x + 8)}{x^{2}} \cdot \frac{x \cdot x^{2} + x (3 x) - 4 x}{x^{2} - 1}$

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

$\frac{(x + 1) (x + 8)}{x^{2}} \cdot \frac{x \cdot x^{2} + x (3 x) + x \cdot - 4}{x^{2} - 1}$

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

$\frac{(x + 1) (x + 8)}{x^{2}} \cdot \frac{x (x^{2} + 3 x) + x \cdot - 4}{x^{2} - 1}$

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

$\frac{(x + 1) (x + 8)}{x^{2}} \cdot \frac{x (x^{2} + 3 x - 4)}{x^{2} - 1}$

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.

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

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

Simplify the denominator.

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

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

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 $b = 1$ .

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

Simplify terms.

Combine.

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

Cancel the common factor of $x + 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x$ and $x^{2}$ .

Factor $x$ out of $(x + 8) (x (x - 1) (x + 4))$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 1$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{(x + 8) (x + 4)}{x}$

Simplify ((x^2+9x+8)/(x^2))÷((x^2-1)/(x^3+3x^2-4x))

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