Algebra Examples

$\frac{2 x^{2} + x - 6}{x + 4} \cdot \frac{x^{2} - 4 x - 32}{x + 2}$

Step 1

Factor by grouping.

Tap for more steps...

Step 1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 6 = - 12$ and whose sum is $b = 1$ .

Tap for more steps...

Step 1.1.1

Multiply by $1$ .

$\frac{2 x^{2} + 1 x - 6}{x + 4} \cdot \frac{x^{2} - 4 x - 32}{x + 2}$

Step 1.1.2

Rewrite $1$ as $- 3$ plus $4$

$\frac{2 x^{2} + (- 3 + 4) x - 6}{x + 4} \cdot \frac{x^{2} - 4 x - 32}{x + 2}$

Step 1.1.3

Apply the distributive property.

$\frac{2 x^{2} - 3 x + 4 x - 6}{x + 4} \cdot \frac{x^{2} - 4 x - 32}{x + 2}$

Step 1.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.2.1

Group the first two terms and the last two terms.

$\frac{(2 x^{2} - 3 x) + 4 x - 6}{x + 4} \cdot \frac{x^{2} - 4 x - 32}{x + 2}$

Step 1.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{x (2 x - 3) + 2 (2 x - 3)}{x + 4} \cdot \frac{x^{2} - 4 x - 32}{x + 2}$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 3$ .

$\frac{(2 x - 3) (x + 2)}{x + 4} \cdot \frac{x^{2} - 4 x - 32}{x + 2}$

Step 2

Factor $x^{2} - 4 x - 32$ using the AC method.

Tap for more steps...

Step 2.1

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 $- 32$ and whose sum is $- 4$ .

$- 8, 4$

Step 2.2

Write the factored form using these integers.

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

Step 3

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 3.1

Cancel the common factor of $x + 2$ .

Tap for more steps...

Step 3.1.1

Factor $x + 2$ out of $(2 x - 3) (x + 2)$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x + 4$ .

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

$(2 x - 3) \cdot (x - 8)$

Step 4

Expand $(2 x - 3) (x - 8)$ using the FOIL Method.

Tap for more steps...

Step 4.1

Apply the distributive property.

$2 x (x - 8) - 3 (x - 8)$

Step 4.2

Apply the distributive property.

$2 x \cdot x + 2 x \cdot - 8 - 3 (x - 8)$

Step 4.3

Apply the distributive property.

$2 x \cdot x + 2 x \cdot - 8 - 3 x - 3 \cdot - 8$

Step 5

Simplify and combine like terms.

Tap for more steps...

Step 5.1

Simplify each term.

Tap for more steps...

Step 5.1.1

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

Tap for more steps...

Step 5.1.1.1

Move $x$ .

$2 (x \cdot x) + 2 x \cdot - 8 - 3 x - 3 \cdot - 8$

Step 5.1.1.2

Multiply $x$ by $x$ .

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

Step 5.1.2

Multiply $- 8$ by $2$ .

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

Step 5.1.3

Multiply $- 3$ by $- 8$ .

$2 x^{2} - 16 x - 3 x + 24$

Step 5.2

Subtract $3 x$ from $- 16 x$ .

$2 x^{2} - 19 x + 24$