Simplify (8x)/(x^2+8x+16)-6/(x^2+4x)

$\frac{8 x}{x^{2} + 8 x + 16} - \frac{6}{x^{2} + 4 x}$

Simplify each term.

Factor using the perfect square rule.

Rewrite $16$ as $4^{2}$ .

$\frac{8 x}{x^{2} + 8 x + 4^{2}} - \frac{6}{x^{2} + 4 x}$

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

$8 x = 2 \cdot x \cdot 4$

Rewrite the polynomial.

$\frac{8 x}{x^{2} + 2 \cdot x \cdot 4 + 4^{2}} - \frac{6}{x^{2} + 4 x}$

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

$\frac{8 x}{{(x + 4)}^{2}} - \frac{6}{x^{2} + 4 x}$

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

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

$\frac{8 x}{{(x + 4)}^{2}} - \frac{6}{x \cdot x + 4 x}$

Factor $x$ out of $4 x$ .

$\frac{8 x}{{(x + 4)}^{2}} - \frac{6}{x \cdot x + x \cdot 4}$

Factor $x$ out of $x \cdot x + x \cdot 4$ .

$\frac{8 x}{{(x + 4)}^{2}} - \frac{6}{x (x + 4)}$

To write $\frac{8 x}{{(x + 4)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{8 x}{{(x + 4)}^{2}} \cdot \frac{x}{x} - \frac{6}{x (x + 4)}$

To write $- \frac{6}{x (x + 4)}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$\frac{8 x}{{(x + 4)}^{2}} \cdot \frac{x}{x} - \frac{6}{x (x + 4)} \cdot \frac{x + 4}{x + 4}$

Write each expression with a common denominator of $x {(x + 4)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{8 x}{{(x + 4)}^{2}}$ by $\frac{x}{x}$ .

$\frac{8 x \cdot x}{{(x + 4)}^{2} x} - \frac{6}{x (x + 4)} \cdot \frac{x + 4}{x + 4}$

Multiply $\frac{6}{x (x + 4)}$ by $\frac{x + 4}{x + 4}$ .

$\frac{8 x \cdot x}{{(x + 4)}^{2} x} - \frac{6 (x + 4)}{x (x + 4) (x + 4)}$

Raise $x + 4$ to the power of $1$ .

$\frac{8 x \cdot x}{{(x + 4)}^{2} x} - \frac{6 (x + 4)}{x ({(x + 4)}^{1} (x + 4))}$

Raise $x + 4$ to the power of $1$ .

$\frac{8 x \cdot x}{{(x + 4)}^{2} x} - \frac{6 (x + 4)}{x ({(x + 4)}^{1} {(x + 4)}^{1})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{8 x \cdot x}{{(x + 4)}^{2} x} - \frac{6 (x + 4)}{x {(x + 4)}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{8 x \cdot x}{{(x + 4)}^{2} x} - \frac{6 (x + 4)}{x {(x + 4)}^{2}}$

Reorder the factors of ${(x + 4)}^{2} x$ .

$\frac{8 x \cdot x}{x {(x + 4)}^{2}} - \frac{6 (x + 4)}{x {(x + 4)}^{2}}$

Combine the numerators over the common denominator.

$\frac{8 x \cdot x - 6 (x + 4)}{x {(x + 4)}^{2}}$

Simplify the numerator.

Factor $2$ out of $8 x \cdot x - 6 (x + 4)$ .

Factor $2$ out of $8 x \cdot x$ .

$\frac{2 (4 x \cdot x) - 6 (x + 4)}{x {(x + 4)}^{2}}$

Factor $2$ out of $- 6 (x + 4)$ .

$\frac{2 (4 x \cdot x) + 2 (- 3 (x + 4))}{x {(x + 4)}^{2}}$

Factor $2$ out of $2 (4 x \cdot x) + 2 (- 3 (x + 4))$ .

$\frac{2 (4 x \cdot x - 3 (x + 4))}{x {(x + 4)}^{2}}$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{2 (4 (x \cdot x) - 3 (x + 4))}{x {(x + 4)}^{2}}$

Multiply $x$ by $x$ .

$\frac{2 (4 x^{2} - 3 (x + 4))}{x {(x + 4)}^{2}}$

Apply the distributive property.

$\frac{2 (4 x^{2} - 3 x - 3 \cdot 4)}{x {(x + 4)}^{2}}$

Multiply $- 3$ by $4$ .

$\frac{2 (4 x^{2} - 3 x - 12)}{x {(x + 4)}^{2}}$

Simplify (8x)/(x^2+8x+16)-6/(x^2+4x)

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