Simplify (11x^2-11m^2)/(2x^2+4xm+2m^2)

$\frac{11 x^{2} - 11 m^{2}}{2 x^{2} + 4 x m + 2 m^{2}}$

Step 1

Simplify the numerator.

Factor $11$ out of $11 x^{2} - 11 m^{2}$ .

Factor $11$ out of $11 x^{2}$ .

$\frac{11 (x^{2}) - 11 m^{2}}{2 x^{2} + 4 x m + 2 m^{2}}$

Factor $11$ out of $- 11 m^{2}$ .

$\frac{11 (x^{2}) + 11 (- m^{2})}{2 x^{2} + 4 x m + 2 m^{2}}$

Factor $11$ out of $11 (x^{2}) + 11 (- m^{2})$ .

$\frac{11 (x^{2} - m^{2})}{2 x^{2} + 4 x m + 2 m^{2}}$

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 $b = m$ .

$\frac{11 (x + m) (x - m)}{2 x^{2} + 4 x m + 2 m^{2}}$

Step 2

Simplify the denominator.

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

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

$\frac{11 (x + m) (x - m)}{2 (x^{2}) + 4 x m + 2 m^{2}}$

Factor $2$ out of $4 x m$ .

$\frac{11 (x + m) (x - m)}{2 (x^{2}) + 2 (2 x m) + 2 m^{2}}$

Factor $2$ out of $2 m^{2}$ .

$\frac{11 (x + m) (x - m)}{2 (x^{2}) + 2 (2 x m) + 2 (m^{2})}$

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

$\frac{11 (x + m) (x - m)}{2 (x^{2} + 2 x m) + 2 (m^{2})}$

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

$\frac{11 (x + m) (x - m)}{2 (x^{2} + 2 x m + m^{2})}$

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.

$2 x m = 2 \cdot x \cdot m$

Rewrite the polynomial.

$\frac{11 (x + m) (x - m)}{2 (x^{2} + 2 \cdot x \cdot m + m^{2})}$

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

$\frac{11 (x + m) (x - m)}{2 {(x + m)}^{2}}$

Step 3

Cancel the common factor of $x + m$ and ${(x + m)}^{2}$ .

Factor $x + m$ out of $11 (x + m) (x - m)$ .

$\frac{(x + m) (11 (x - m))}{2 {(x + m)}^{2}}$

Cancel the common factors.

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

$\frac{(x + m) (11 (x - m))}{(x + m) (2 (x + m))}$

Cancel the common factor.

$\frac{(x + m) (11 (x - m))}{(x + m) (2 (x + m))}$

Rewrite the expression.

$\frac{11 (x - m)}{2 (x + m)}$