Simplify (25-16x^2)/(16x^2+40x+25)

$\frac{25 - 16 x^{2}}{16 x^{2} + 40 x + 25}$

Simplify the numerator.

Rewrite $25$ as $5^{2}$ .

$\frac{5^{2} - 16 x^{2}}{16 x^{2} + 40 x + 25}$

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

$\frac{5^{2} - {(4 x)}^{2}}{16 x^{2} + 40 x + 25}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = 4 x$ .

$\frac{(5 + 4 x) (5 - (4 x))}{16 x^{2} + 40 x + 25}$

Multiply $4$ by $- 1$ .

$\frac{(5 + 4 x) (5 - 4 x)}{16 x^{2} + 40 x + 25}$

Factor using the perfect square rule.

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

$\frac{(5 + 4 x) (5 - 4 x)}{{(4 x)}^{2} + 40 x + 25}$

Rewrite $25$ as $5^{2}$ .

$\frac{(5 + 4 x) (5 - 4 x)}{{(4 x)}^{2} + 40 x + 5^{2}}$

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

$40 x = 2 \cdot (4 x) \cdot 5$

Rewrite the polynomial.

$\frac{(5 + 4 x) (5 - 4 x)}{{(4 x)}^{2} + 2 \cdot (4 x) \cdot 5 + 5^{2}}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 4 x$ and $b = 5$ .

$\frac{(5 + 4 x) (5 - 4 x)}{{(4 x + 5)}^{2}}$

Cancel the common factor of $5 + 4 x$ and ${(4 x + 5)}^{2}$ .

Reorder terms.

$\frac{(4 x + 5) (5 - 4 x)}{{(4 x + 5)}^{2}}$

Cancel the common factors.

Factor $4 x + 5$ out of ${(4 x + 5)}^{2}$ .

$\frac{(4 x + 5) (5 - 4 x)}{(4 x + 5) (4 x + 5)}$

Cancel the common factor.

$\frac{(4 x + 5) (5 - 4 x)}{(4 x + 5) (4 x + 5)}$

Rewrite the expression.

$\frac{5 - 4 x}{4 x + 5}$