Algebra Examples

$\frac{{(x - y)}^{2}}{2 x y + 6 y} \cdot \frac{4 x + 12}{x^{2} - y^{2}}$

Step 1

Simplify with factoring out.

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Step 1.1

Factor $2 y$ out of $2 x y + 6 y$ .

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Step 1.1.1

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

$\frac{{(x - y)}^{2}}{2 y (x) + 6 y} \cdot \frac{4 x + 12}{x^{2} - y^{2}}$

Step 1.1.2

Factor $2 y$ out of $6 y$ .

$\frac{{(x - y)}^{2}}{2 y (x) + 2 y (3)} \cdot \frac{4 x + 12}{x^{2} - y^{2}}$

Step 1.1.3

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

$\frac{{(x - y)}^{2}}{2 y (x + 3)} \cdot \frac{4 x + 12}{x^{2} - y^{2}}$

Step 1.2

Factor $4$ out of $4 x + 12$ .

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Step 1.2.1

Factor $4$ out of $4 x$ .

$\frac{{(x - y)}^{2}}{2 y (x + 3)} \cdot \frac{4 (x) + 12}{x^{2} - y^{2}}$

Step 1.2.2

Factor $4$ out of $12$ .

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

Step 1.2.3

Factor $4$ out of $4 x + 4 \cdot 3$ .

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

Combine.

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

Step 3.2

Cancel the common factor of ${(x - y)}^{2}$ and $x - y$ .

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Step 3.2.1

Factor $x - y$ out of ${(x - y)}^{2} (4 (x + 3))$ .

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

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $x - y$ out of $2 y (x + 3) ((x + y) (x - y))$ .

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $4$ and $2$ .

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

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

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

Step 3.3.2.2

Cancel the common factor.

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

Step 3.3.2.3

Rewrite the expression.

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

Step 3.4

Cancel the common factor of $x + 3$ .

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Rewrite the expression.

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

Step 3.5

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

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