Simplify (2x^2+5x+2)/(4x^2-1)*(2x^2+x-1)/(x^2+x-2)

$\frac{2 x^{2} + 5 x + 2}{4 x^{2} - 1} \cdot \frac{2 x^{2} + x - 1}{x^{2} + x - 2}$

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 $a \cdot c = 2 \cdot 2 = 4$ and whose sum is $b = 5$ .

Factor $5$ out of $5 x$ .

$\frac{2 x^{2} + 5 (x) + 2}{4 x^{2} - 1} \cdot \frac{2 x^{2} + x - 1}{x^{2} + x - 2}$

Rewrite $5$ as $1$ plus $4$

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

Apply the distributive property.

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

Multiply $x$ by $1$ .

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

Step 2

Simplify the denominator.

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

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

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

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 1$ .

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

Step 3

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 $b = 1$ .

Multiply by $1$ .

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

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

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

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

Step 4

Factor $x^{2} + x - 2$ 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 $1$ .

$- 1, 2$

Write the factored form using these integers.

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

Step 5

Simplify terms.

Cancel the common factor of $x + 2$ .

Factor $x + 2$ out of $(2 x + 1) (x + 2)$ .

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

Factor $x + 2$ out of $(x - 1) (x + 2)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2 x - 1$ .

Factor $2 x - 1$ out of $(2 x + 1) (2 x - 1)$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{2 x + 1}{2 x + 1} \cdot \frac{x + 1}{x - 1}$

Multiply $\frac{2 x + 1}{2 x + 1}$ by $\frac{x + 1}{x - 1}$ .

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

Cancel the common factor of $2 x + 1$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{x + 1}{x - 1}$

Simplify (2x^2+5x+2)/(4x^2-1)*(2x^2+x-1)/(x^2+x-2)

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