Solve frac{a^{2}+3a+2}{a^{2}+2a+4}div(a+1)^2/a^3-8*a^2+4a+3/a^2+a-2

$$\frac { a ^ { 2 } + 3 a + 2 } { a ^ { 2 } + 2 a + 4 } \div \frac { ( a + 1 ) ^ { 2 } } { a ^ { 3 } - 8 } \times \frac { a ^ { 2 } + 4 a + 3 } { a ^ { 2 } + a - 2 }$$

Evaluate

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

Short Solution Steps

Divide \frac{a^{2}+3a+2}{a^{2}+2a+4} by \frac{\left(a+1\right)^{2}}{a^{3}-8} by multiplying \frac{a^{2}+3a+2}{a^{2}+2a+4} by the reciprocal of \frac{\left(a+1\right)^{2}}{a^{3}-8}.

$$\frac{\left(a^{2}+3a+2\right)\left(a^{3}-8\right)}{\left(a^{2}+2a+4\right)\left(a+1\right)^{2}}\times \frac{a^{2}+4a+3}{a^{2}+a-2}$$

Factor the expressions that are not already factored in \frac{\left(a^{2}+3a+2\right)\left(a^{3}-8\right)}{\left(a^{2}+2a+4\right)\left(a+1\right)^{2}}.

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

Cancel out \left(a+1\right)\left(a^{2}+2a+4\right) in both numerator and denominator.

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

Multiply \frac{\left(a-2\right)\left(a+2\right)}{a+1} times \frac{a^{2}+4a+3}{a^{2}+a-2} by multiplying numerator times numerator and denominator times denominator.

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

Factor the expressions that are not already factored.

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

Cancel out \left(a+1\right)\left(a+2\right) in both numerator and denominator.

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

Expand the expression.

$$\frac{a^{2}+a-6}{a-1}$$

Expand

$$\frac{a^{2}+a-6}{a-1}$$

Short Solution Steps

Divide \frac{a^{2}+3a+2}{a^{2}+2a+4} by \frac{\left(a+1\right)^{2}}{a^{3}-8} by multiplying \frac{a^{2}+3a+2}{a^{2}+2a+4} by the reciprocal of \frac{\left(a+1\right)^{2}}{a^{3}-8}.

$$\frac{\left(a^{2}+3a+2\right)\left(a^{3}-8\right)}{\left(a^{2}+2a+4\right)\left(a+1\right)^{2}}\times \frac{a^{2}+4a+3}{a^{2}+a-2}$$

Factor the expressions that are not already factored in \frac{\left(a^{2}+3a+2\right)\left(a^{3}-8\right)}{\left(a^{2}+2a+4\right)\left(a+1\right)^{2}}.

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

Cancel out \left(a+1\right)\left(a^{2}+2a+4\right) in both numerator and denominator.

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

Multiply \frac{\left(a-2\right)\left(a+2\right)}{a+1} times \frac{a^{2}+4a+3}{a^{2}+a-2} by multiplying numerator times numerator and denominator times denominator.

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

Factor the expressions that are not already factored.

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

Cancel out \left(a+1\right)\left(a+2\right) in both numerator and denominator.

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

Expand the expression.

$$\frac{a^{2}+a-6}{a-1}$$