Divide ((p^2-p-2)/(2p^2+3p+1))÷((p^2-3p+2)/(p^2+7p-8))

$\frac{p^{2} - p - 2}{2 p^{2} + 3 p + 1} \div \frac{p^{2} - 3 p + 2}{p^{2} + 7 p - 8}$

To divide by a fraction, multiply by its reciprocal.

$\frac{p^{2} - p - 2}{2 p^{2} + 3 p + 1} \cdot \frac{p^{2} + 7 p - 8}{p^{2} - 3 p + 2}$

Factor $p^{2} - p - 2$ 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 $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Write the factored form using these integers.

$\frac{(p - 2) (p + 1)}{2 p^{2} + 3 p + 1} \cdot \frac{p^{2} + 7 p - 8}{p^{2} - 3 p + 2}$

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

Factor $3$ out of $3 p$ .

$\frac{(p - 2) (p + 1)}{2 p^{2} + 3 (p) + 1} \cdot \frac{p^{2} + 7 p - 8}{p^{2} - 3 p + 2}$

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

$\frac{(p - 2) (p + 1)}{2 p^{2} + (1 + 2) p + 1} \cdot \frac{p^{2} + 7 p - 8}{p^{2} - 3 p + 2}$

Apply the distributive property.

$\frac{(p - 2) (p + 1)}{2 p^{2} + 1 p + 2 p + 1} \cdot \frac{p^{2} + 7 p - 8}{p^{2} - 3 p + 2}$

Multiply $p$ by $1$ .

$\frac{(p - 2) (p + 1)}{2 p^{2} + p + 2 p + 1} \cdot \frac{p^{2} + 7 p - 8}{p^{2} - 3 p + 2}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(p - 2) (p + 1)}{(2 p^{2} + p) + 2 p + 1} \cdot \frac{p^{2} + 7 p - 8}{p^{2} - 3 p + 2}$

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

$\frac{(p - 2) (p + 1)}{p (2 p + 1) + 1 (2 p + 1)} \cdot \frac{p^{2} + 7 p - 8}{p^{2} - 3 p + 2}$

Factor the polynomial by factoring out the greatest common factor, $2 p + 1$ .

$\frac{(p - 2) (p + 1)}{(2 p + 1) (p + 1)} \cdot \frac{p^{2} + 7 p - 8}{p^{2} - 3 p + 2}$

Cancel the common factor of $p + 1$ .

Cancel the common factor.

$\frac{(p - 2) (p + 1)}{(2 p + 1) (p + 1)} \cdot \frac{p^{2} + 7 p - 8}{p^{2} - 3 p + 2}$

Rewrite the expression.

$\frac{p - 2}{2 p + 1} \cdot \frac{p^{2} + 7 p - 8}{p^{2} - 3 p + 2}$

Factor $p^{2} + 7 p - 8$ 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 $7$ .

$- 1, 8$

Write the factored form using these integers.

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

Factor $p^{2} - 3 p + 2$ 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 $2$ and whose sum is $- 3$ .

$- 2, - 1$

Write the factored form using these integers.

$\frac{p - 2}{2 p + 1} \cdot \frac{(p - 1) (p + 8)}{(p - 2) (p - 1)}$

Cancel the common factor of $p - 2$ .

Cancel the common factor.

$\frac{p - 2}{2 p + 1} \cdot \frac{(p - 1) (p + 8)}{(p - 2) (p - 1)}$

Rewrite the expression.

$\frac{1}{2 p + 1} \cdot \frac{(p - 1) (p + 8)}{p - 1}$

Multiply $\frac{1}{2 p + 1}$ by $\frac{(p - 1) (p + 8)}{p - 1}$ .

$\frac{(p - 1) (p + 8)}{(2 p + 1) (p - 1)}$

Cancel the common factor of $p - 1$ .

Cancel the common factor.

$\frac{(p - 1) (p + 8)}{(2 p + 1) (p - 1)}$

Rewrite the expression.

$\frac{p + 8}{2 p + 1}$

Divide ((p^2-p-2)/(2p^2+3p+1))÷((p^2-3p+2)/(p^2+7p-8))

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