Solve x^2-16/x^2+5x+6divx^2+5x+4/x^2-2x-8

$$\frac{ { x }^{ 2 } -16 }{ { x }^{ 2 } +5x+6 } \div \frac{ { x }^{ 2 } +5x+4 }{ { x }^{ 2 } -2x-8 }$$

Evaluate

$$\frac{\left(x-4\right)^{2}}{\left(x+1\right)\left(x+3\right)}$$

Short Solution Steps

Divide \frac{x^{2}-16}{x^{2}+5x+6} by \frac{x^{2}+5x+4}{x^{2}-2x-8} by multiplying \frac{x^{2}-16}{x^{2}+5x+6} by the reciprocal of \frac{x^{2}+5x+4}{x^{2}-2x-8}.

$$\frac{\left(x^{2}-16\right)\left(x^{2}-2x-8\right)}{\left(x^{2}+5x+6\right)\left(x^{2}+5x+4\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(x+2\right)\left(x+4\right)\left(x-4\right)^{2}}{\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)}$$

Cancel out \left(x+2\right)\left(x+4\right) in both numerator and denominator.

$$\frac{\left(x-4\right)^{2}}{\left(x+1\right)\left(x+3\right)}$$

Expand the expression.

$$\frac{x^{2}-8x+16}{x^{2}+4x+3}$$

Expand

$$\frac{x^{2}-8x+16}{\left(x+1\right)\left(x+3\right)}$$

Short Solution Steps

Divide \frac{x^{2}-16}{x^{2}+5x+6} by \frac{x^{2}+5x+4}{x^{2}-2x-8} by multiplying \frac{x^{2}-16}{x^{2}+5x+6} by the reciprocal of \frac{x^{2}+5x+4}{x^{2}-2x-8}.

$$\frac{\left(x^{2}-16\right)\left(x^{2}-2x-8\right)}{\left(x^{2}+5x+6\right)\left(x^{2}+5x+4\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(x+2\right)\left(x+4\right)\left(x-4\right)^{2}}{\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)}$$

Cancel out \left(x+2\right)\left(x+4\right) in both numerator and denominator.

$$\frac{\left(x-4\right)^{2}}{\left(x+1\right)\left(x+3\right)}$$

Expand the expression.

$$\frac{x^{2}-8x+16}{x^{2}+4x+3}$$