Find the Product (2x^2-x-3)/(x+4)*(x^2-2x-24)/(x+1)

$\frac{2 x^{2} - x - 3}{x + 4} \cdot \frac{x^{2} - 2 x - 24}{x + 1}$

Step 1

Factor by grouping.

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 $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = - 1$ .

Factor $- 1$ out of $- x$ .

$\frac{2 x^{2} - (x) - 3}{x + 4} \cdot \frac{x^{2} - 2 x - 24}{x + 1}$

Rewrite $- 1$ as $2$ plus $- 3$

$\frac{2 x^{2} + (2 - 3) x - 3}{x + 4} \cdot \frac{x^{2} - 2 x - 24}{x + 1}$

Apply the distributive property.

$\frac{2 x^{2} + 2 x - 3 x - 3}{x + 4} \cdot \frac{x^{2} - 2 x - 24}{x + 1}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(2 x^{2} + 2 x) - 3 x - 3}{x + 4} \cdot \frac{x^{2} - 2 x - 24}{x + 1}$

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

$\frac{2 x (x + 1) - 3 (x + 1)}{x + 4} \cdot \frac{x^{2} - 2 x - 24}{x + 1}$

Factor the polynomial by factoring out the greatest common factor, $x + 1$ .

$\frac{(x + 1) (2 x - 3)}{x + 4} \cdot \frac{x^{2} - 2 x - 24}{x + 1}$

Step 2

Factor $x^{2} - 2 x - 24$ 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 $- 24$ and whose sum is $- 2$ .

$- 6, 4$

Write the factored form using these integers.

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

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x + 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 4$ .

Factor $x + 4$ out of $(x - 6) (x + 4)$ .

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

Cancel the common factor.

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

Rewrite the expression.

$(2 x - 3) \cdot (x - 6)$

Step 4

Expand $(2 x - 3) (x - 6)$ using the FOIL Method.

Apply the distributive property.

$2 x (x - 6) - 3 (x - 6)$

Apply the distributive property.

$2 x \cdot x + 2 x \cdot - 6 - 3 (x - 6)$

Apply the distributive property.

$2 x \cdot x + 2 x \cdot - 6 - 3 x - 3 \cdot - 6$

Step 5

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$2 (x \cdot x) + 2 x \cdot - 6 - 3 x - 3 \cdot - 6$

Multiply $x$ by $x$ .

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

Multiply $- 6$ by $2$ .

$2 x^{2} - 12 x - 3 x - 3 \cdot - 6$

Multiply $- 3$ by $- 6$ .

$2 x^{2} - 12 x - 3 x + 18$

Subtract $3 x$ from $- 12 x$ .

$2 x^{2} - 15 x + 18$

Find the Product (2x^2-x-3)/(x+4)*(x^2-2x-24)/(x+1)

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