Solve (x+1)/((x)^(2)+4x+3)\times((x)^(2)+3x+2)/(x-2)\div(x+1)/(x+3)

Question

$$\frac{ x+1 }{ { x }^{ 2 } +4x+3 } \times \frac{ { x }^{ 2 } +3x+2 }{ x-2 } \div \frac{ x+1 }{ x+3 }$$

Evaluate

$\frac{x+2}{x-2}$

Short Solution Steps

Factor the expressions that are not already factored in $\frac{x+1}{x^{2}+4x+3}$.

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

Cancel out $x+1$ in both numerator and denominator.

$$\frac{\frac{1}{x+3}\times \frac{x^{2}+3x+2}{x-2}}{\frac{x+1}{x+3}}$$

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

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

Divide $\frac{x^{2}+3x+2}{\left(x+3\right)\left(x-2\right)}$ by $\frac{x+1}{x+3}$ by multiplying $\frac{x^{2}+3x+2}{\left(x+3\right)\left(x-2\right)}$ by the reciprocal of $\frac{x+1}{x+3}$.

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

Cancel out $x+3$ in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out $x+1$ in both numerator and denominator.

$$\frac{x+2}{x-2}$$

Show Solution

Expand

$\frac{x+2}{x-2}$

Short Solution Steps

Factor the expressions that are not already factored in $\frac{x+1}{x^{2}+4x+3}$.

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

Cancel out $x+1$ in both numerator and denominator.

$$\frac{\frac{1}{x+3}\times \frac{x^{2}+3x+2}{x-2}}{\frac{x+1}{x+3}}$$

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

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

Divide $\frac{x^{2}+3x+2}{\left(x+3\right)\left(x-2\right)}$ by $\frac{x+1}{x+3}$ by multiplying $\frac{x^{2}+3x+2}{\left(x+3\right)\left(x-2\right)}$ by the reciprocal of $\frac{x+1}{x+3}$.

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

Cancel out $x+3$ in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out $x+1$ in both numerator and denominator.

$$\frac{x+2}{x-2}$$

Show Solution

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