Simplify (x^2+xy-2y^2)/(x^3+x^2y)*x/(x^2+3xy+2y^2)

$\frac{x^{2} + x y - 2 y^{2}}{x^{3} + x^{2} y} \cdot \frac{x}{x^{2} + 3 x y + 2 y^{2}}$

Step 1

Simplify the numerator.

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

Reorder terms.

$\frac{x^{2} - 2 y^{2} + x y}{x^{3} + x^{2} y} \cdot \frac{x}{x^{2} + 3 x y + 2 y^{2}}$

Reorder $- 2 y^{2}$ and $x y$ .

$\frac{x^{2} + x y - 2 y^{2}}{x^{3} + x^{2} y} \cdot \frac{x}{x^{2} + 3 x y + 2 y^{2}}$

Factor $1$ out of $x y$ .

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

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

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

Apply the distributive property.

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

Move parentheses.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

Rewrite $- 1 y$ as $- y$ .

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

Step 2

Factor $x^{2}$ out of $x^{3} + x^{2} y$ .

Factor $x^{2}$ out of $x^{3}$ .

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

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

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

Factor $x^{2}$ out of $x^{2} x + x^{2} (y)$ .

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

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

Reorder terms.

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

Reorder $2 y^{2}$ and $3 x y$ .

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

Factor $3$ out of $3 x y$ .

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

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

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

Apply the distributive property.

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

Multiply $x y$ by $1$ .

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

Move parentheses.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

Step 4

Simplify terms.

Combine.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x$ and $x^{2}$ .

Factor $x$ out of $(x - y) x$ .

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

Cancel the common factors.

Factor $x$ out of $x^{2} (x + y) (x + y)$ .

$\frac{x (x - y)}{x ((x (x + y)) (x + y))}$

Cancel the common factor.

$\frac{x (x - y)}{x ((x (x + y)) (x + y))}$

Rewrite the expression.

$\frac{x - y}{(x (x + y)) (x + y)}$

Step 5

Simplify the denominator.

Raise $x + y$ to the power of $1$ .

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

Raise $x + y$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

Simplify (x^2+xy-2y^2)/(x^3+x^2y)*x/(x^2+3xy+2y^2)

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