Simplify ((a^2+ax+x^2)/(ax+2ay))÷((a^3-x^3)/(bx+2by))

$\frac{a^{2} + a x + x^{2}}{a x + 2 a y} \div \frac{a^{3} - x^{3}}{b x + 2 b y}$

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Simplify with factoring out.

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

Factor $a$ out of $a x$ .

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

Factor $a$ out of $2 a y$ .

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

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

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

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

Factor $b$ out of $b x$ .

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

Factor $b$ out of $2 b y$ .

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

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

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

Step 3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = a$ and $b = x$ .

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

Step 4

Simplify terms.

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{a} \cdot \frac{b}{a - x}$

Multiply $\frac{1}{a}$ by $\frac{b}{a - x}$ .

$\frac{b}{a (a - x)}$

Simplify ((a^2+ax+x^2)/(ax+2ay))÷((a^3-x^3)/(bx+2by))

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