Simplify (x^2-2x-8)/(9x)*(3x^3)/(x^2-4x)

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

Step 1

Factor $x^{2} - 2 x - 8$ 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 $- 2$ .

$- 4, 2$

Write the factored form using these integers.

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

Step 2

Simplify terms.

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

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

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

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

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

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

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

Combine.

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

Step 3

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

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

Multiply $x$ by $x$ .

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

Step 4

Cancel the common factor of $x - 4$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{(x + 2) (3 x^{3})}{9 x^{2}}$

Step 5

Cancel the common factor of $3$ and $9$ .

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

$\frac{3 ((x + 2) (x^{3}))}{9 x^{2}}$

Cancel the common factors.

Factor $3$ out of $9 x^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

Cancel the common factor of $x^{3}$ and $x^{2}$ .

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{(x + 2) (x)}{3}$

Step 7

Reorder factors in $\frac{(x + 2) (x)}{3}$ .

$\frac{x (x + 2)}{3}$

Simplify (x^2-2x-8)/(9x)*(3x^3)/(x^2-4x)

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