Simplify (a(a^2-b^2))/(a^2+2ab+b^2)

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

Step 1

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

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

Step 2

Factor using the perfect square rule.

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 a b = 2 \cdot a \cdot b$

Rewrite the polynomial.

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

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = a$ and $b = b$ .

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

Step 3

Cancel the common factor of $a + b$ and ${(a + b)}^{2}$ .

Factor $a + b$ out of $a (a + b) (a - b)$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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