Simplify ((4x^2+2x)/(x^2+x-2))/((8x^2+4x)/(3x^2+10x+8))

$\frac{\frac{4 x^{2} + 2 x}{x^{2} + x - 2}}{\frac{8 x^{2} + 4 x}{3 x^{2} + 10 x + 8}}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{4 x^{2} + 2 x}{x^{2} + x - 2} \cdot \frac{3 x^{2} + 10 x + 8}{8 x^{2} + 4 x}$

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

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

$\frac{2 x (2 x) + 2 x}{x^{2} + x - 2} \cdot \frac{3 x^{2} + 10 x + 8}{8 x^{2} + 4 x}$

Factor $2 x$ out of $2 x$ .

$\frac{2 x (2 x) + 2 x (1)}{x^{2} + x - 2} \cdot \frac{3 x^{2} + 10 x + 8}{8 x^{2} + 4 x}$

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

$\frac{2 x (2 x + 1)}{x^{2} + x - 2} \cdot \frac{3 x^{2} + 10 x + 8}{8 x^{2} + 4 x}$

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.

$\frac{2 x (2 x + 1)}{(x - 1) (x + 2)} \cdot \frac{3 x^{2} + 10 x + 8}{8 x^{2} + 4 x}$

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 $a \cdot c = 3 \cdot 8 = 24$ and whose sum is $b = 10$ .

Factor $10$ out of $10 x$ .

$\frac{2 x (2 x + 1)}{(x - 1) (x + 2)} \cdot \frac{3 x^{2} + 10 (x) + 8}{8 x^{2} + 4 x}$

Rewrite $10$ as $4$ plus $6$

$\frac{2 x (2 x + 1)}{(x - 1) (x + 2)} \cdot \frac{3 x^{2} + (4 + 6) x + 8}{8 x^{2} + 4 x}$

Apply the distributive property.

$\frac{2 x (2 x + 1)}{(x - 1) (x + 2)} \cdot \frac{3 x^{2} + 4 x + 6 x + 8}{8 x^{2} + 4 x}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{2 x (2 x + 1)}{(x - 1) (x + 2)} \cdot \frac{(3 x^{2} + 4 x) + 6 x + 8}{8 x^{2} + 4 x}$

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

$\frac{2 x (2 x + 1)}{(x - 1) (x + 2)} \cdot \frac{x (3 x + 4) + 2 (3 x + 4)}{8 x^{2} + 4 x}$

Factor the polynomial by factoring out the greatest common factor, $3 x + 4$ .

$\frac{2 x (2 x + 1)}{(x - 1) (x + 2)} \cdot \frac{(3 x + 4) (x + 2)}{8 x^{2} + 4 x}$

Factor $4 x$ out of $8 x^{2} + 4 x$ .

Factor $4 x$ out of $8 x^{2}$ .

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

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

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

Factor $4 x$ out of $4 x (2 x) + 4 x (1)$ .

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

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

Factor $2 x (2 x + 1)$ out of $4 x (2 x + 1)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 2$ .

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

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

Factor $x + 2$ out of $(3 x + 4) (x + 2)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{1}{x - 1}$ by $\frac{3 x + 4}{2}$ .

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

Move $2$ to the left of $x - 1$ .

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

Simplify ((4x^2+2x)/(x^2+x-2))/((8x^2+4x)/(3x^2+10x+8))

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