Solve ((x)^(2)-4)/(x+3)\div(x-2)/((x)^(2)+x-2)\times(x+3)/(x+2)

Question

$$\frac{ { x }^{ 2 } -4 }{ x+3 } \div \frac{ x-2 }{ { x }^{ 2 } +x-2 } \times \frac{ x+3 }{ x+2 }$$

Evaluate

$\left(x-1\right)\left(x+2\right)$

Short Solution Steps

Divide $\frac{x^{2}-4}{x+3}$ by $\frac{x-2}{x^{2}+x-2}$ by multiplying $\frac{x^{2}-4}{x+3}$ by the reciprocal of $\frac{x-2}{x^{2}+x-2}$.

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

Factor the expressions that are not already factored in $\frac{\left(x^{2}-4\right)\left(x^{2}+x-2\right)}{\left(x+3\right)\left(x-2\right)}$.

$$\frac{\left(x-2\right)\left(x-1\right)\left(x+2\right)^{2}}{\left(x-2\right)\left(x+3\right)}\times \frac{x+3}{x+2}$$

Cancel out $x-2$ in both numerator and denominator.

$$\frac{\left(x-1\right)\left(x+2\right)^{2}}{x+3}\times \frac{x+3}{x+2}$$

Multiply $\frac{\left(x-1\right)\left(x+2\right)^{2}}{x+3}$ times $\frac{x+3}{x+2}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x-1\right)\left(x+2\right)^{2}\left(x+3\right)}{\left(x+3\right)\left(x+2\right)}$$

Cancel out $\left(x+2\right)\left(x+3\right)$ in both numerator and denominator.

$$\left(x-1\right)\left(x+2\right)$$

Use the distributive property to multiply $x-1$ by $x+2$ and combine like terms.

$$x^{2}+x-2$$

Show Solution

Expand

$x^{2}+x-2$

Short Solution Steps

Divide $\frac{x^{2}-4}{x+3}$ by $\frac{x-2}{x^{2}+x-2}$ by multiplying $\frac{x^{2}-4}{x+3}$ by the reciprocal of $\frac{x-2}{x^{2}+x-2}$.

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

Factor the expressions that are not already factored in $\frac{\left(x^{2}-4\right)\left(x^{2}+x-2\right)}{\left(x+3\right)\left(x-2\right)}$.

$$\frac{\left(x-2\right)\left(x-1\right)\left(x+2\right)^{2}}{\left(x-2\right)\left(x+3\right)}\times \frac{x+3}{x+2}$$

Cancel out $x-2$ in both numerator and denominator.

$$\frac{\left(x-1\right)\left(x+2\right)^{2}}{x+3}\times \frac{x+3}{x+2}$$

Multiply $\frac{\left(x-1\right)\left(x+2\right)^{2}}{x+3}$ times $\frac{x+3}{x+2}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x-1\right)\left(x+2\right)^{2}\left(x+3\right)}{\left(x+3\right)\left(x+2\right)}$$

Cancel out $\left(x+2\right)\left(x+3\right)$ in both numerator and denominator.

$$\left(x-1\right)\left(x+2\right)$$

Use the distributive property to multiply $x-1$ by $x+2$ and combine like terms.

$$x^{2}+x-2$$

Show Solution

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Short Solution Steps