Expand the Trigonometric Expression (x^2+8xk+16k^2)/(x^2-16k^2)

$\frac{x^{2} + 8 x k + 16 k^{2}}{x^{2} - 16 k^{2}}$

Factor using the perfect square rule.

Rewrite $16 k^{2}$ as ${(4 k)}^{2}$ .

$\frac{x^{2} + 8 x k + {(4 k)}^{2}}{x^{2} - 16 k^{2}}$

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

$8 x k = 2 \cdot x \cdot (4 k)$

Rewrite the polynomial.

$\frac{x^{2} + 2 \cdot x \cdot (4 k) + {(4 k)}^{2}}{x^{2} - 16 k^{2}}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 4 k$ .

$\frac{{(x + 4 k)}^{2}}{x^{2} - 16 k^{2}}$

Simplify the denominator.

Rewrite $16 k^{2}$ as ${(4 k)}^{2}$ .

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

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 $b = 4 k$ .

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

Multiply $4$ by $- 1$ .

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

Cancel the common factor of ${(x + 4 k)}^{2}$ and $x + 4 k$ .

Factor $x + 4 k$ out of ${(x + 4 k)}^{2}$ .

$\frac{(x + 4 k) (x + 4 k)}{(x + 4 k) (x - 4 k)}$

Cancel the common factors.

Cancel the common factor.

$\frac{(x + 4 k) (x + 4 k)}{(x + 4 k) (x - 4 k)}$

Rewrite the expression.

$\frac{x + 4 k}{x - 4 k}$