Simplify ((x^2)/2-2)/(5+10/x)

$\frac{\frac{x^{2}}{2} - 2}{5 + \frac{10}{x}}$

Multiply the numerator and denominator of the complex fraction by $2 x$ .

Multiply $\frac{\frac{x^{2}}{2} - 2}{5 + \frac{10}{x}}$ by $\frac{2 x}{2 x}$ .

$\frac{2 x}{2 x} \cdot \frac{\frac{x^{2}}{2} - 2}{5 + \frac{10}{x}}$

Combine.

$\frac{2 x (\frac{x^{2}}{2} - 2)}{2 x (5 + \frac{10}{x})}$

Apply the distributive property.

$\frac{2 x \frac{x^{2}}{2} + 2 x \cdot - 2}{2 x \cdot 5 + 2 x \frac{10}{x}}$

Simplify by cancelling.

Cancel the common factor of $2$ .

Factor $2$ out of $2 x$ .

$\frac{2 (x) \frac{x^{2}}{2} + 2 x \cdot - 2}{2 x \cdot 5 + 2 x \frac{10}{x}}$

Cancel the common factor.

$\frac{2 x \frac{x^{2}}{2} + 2 x \cdot - 2}{2 x \cdot 5 + 2 x \frac{10}{x}}$

Rewrite the expression.

$\frac{x \cdot x^{2} + 2 x \cdot - 2}{2 x \cdot 5 + 2 x \frac{10}{x}}$

Multiply $x$ by $x^{2}$ by adding the exponents.

Multiply $x$ by $x^{2}$ .

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

$\frac{x^{1} x^{2} + 2 x \cdot - 2}{2 x \cdot 5 + 2 x \frac{10}{x}}$

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

$\frac{x^{1 + 2} + 2 x \cdot - 2}{2 x \cdot 5 + 2 x \frac{10}{x}}$

Add $1$ and $2$ .

$\frac{x^{3} + 2 x \cdot - 2}{2 x \cdot 5 + 2 x \frac{10}{x}}$

Cancel the common factor of $x$ .

Factor $x$ out of $2 x$ .

$\frac{x^{3} + 2 x \cdot - 2}{2 x \cdot 5 + x \cdot 2 \frac{10}{x}}$

Cancel the common factor.

$\frac{x^{3} + 2 x \cdot - 2}{2 x \cdot 5 + x \cdot 2 \frac{10}{x}}$

Rewrite the expression.

$\frac{x^{3} + 2 x \cdot - 2}{2 x \cdot 5 + 2 \cdot 10}$

Multiply $2$ by $10$ .

$\frac{x^{3} + 2 x \cdot - 2}{2 x \cdot 5 + 20}$

Simplify the numerator.

Factor $x$ out of $x^{3} + 2 x \cdot - 2$ .

Factor $x$ out of $x^{3}$ .

$\frac{x \cdot x^{2} + 2 x \cdot - 2}{2 x \cdot 5 + 20}$

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

$\frac{x \cdot x^{2} + x (2 \cdot - 2)}{2 x \cdot 5 + 20}$

Factor $x$ out of $x \cdot x^{2} + x (2 \cdot - 2)$ .

$\frac{x (x^{2} + 2 \cdot - 2)}{2 x \cdot 5 + 20}$

Multiply $2$ by $- 2$ .

$\frac{x (x^{2} - 4)}{2 x \cdot 5 + 20}$

Rewrite $x^{2} - 4$ in a factored form.

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

$\frac{x (x^{2} - 2^{2})}{2 x \cdot 5 + 20}$

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 $b = 2$ .

$\frac{x ((x + 2) (x - 2))}{2 x \cdot 5 + 20}$

$\frac{x (x + 2) (x - 2)}{2 x \cdot 5 + 20}$

Simplify the denominator.

Multiply $5$ by $2$ .

$\frac{x (x + 2) (x - 2)}{10 x + 20}$

Factor $10$ out of $10 x + 20$ .

Factor $10$ out of $10 x$ .

$\frac{x (x + 2) (x - 2)}{10 (x) + 20}$

Factor $10$ out of $20$ .

$\frac{x (x + 2) (x - 2)}{10 x + 10 \cdot 2}$

Factor $10$ out of $10 x + 10 \cdot 2$ .

$\frac{x (x + 2) (x - 2)}{10 (x + 2)}$

Cancel the common factor of $x + 2$ .

Cancel the common factor.

$\frac{x (x + 2) (x - 2)}{10 (x + 2)}$

Rewrite the expression.

$\frac{x (x - 2)}{10}$

Simplify ((x^2)/2-2)/(5+10/x)

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