Solve (a+3/a^{2-1}-a+1/a-1+1)diva^3+8/a^4+3a^3+2a^2

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

Evaluate

$$-\frac{a^{2}}{a^{2}-2a+4}$$

Short Solution Steps

Factor a^{2}-1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-1\right)\left(a+1\right) and a-1 is \left(a-1\right)\left(a+1\right). Multiply \frac{a+1}{a-1} times \frac{a+1}{a+1}.

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

Since \frac{a+3}{\left(a-1\right)\left(a+1\right)} and \frac{\left(a+1\right)\left(a+1\right)}{\left(a-1\right)\left(a+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in a+3-\left(a+1\right)\left(a+1\right).

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

Combine like terms in a+3-a^{2}-a-a-1.

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

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

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

Extract the negative sign in 1-a.

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

Cancel out a-1 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{a+1}{a+1}.

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

Since \frac{-\left(a+2\right)}{a+1} and \frac{a+1}{a+1} have the same denominator, add them by adding their numerators.

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

Do the multiplications in -\left(a+2\right)+a+1.

$$\frac{\frac{-a-2+a+1}{a+1}}{\frac{a^{3}+8}{a^{4}+3a^{3}+2a^{2}}}$$

Combine like terms in -a-2+a+1.

$$\frac{\frac{-1}{a+1}}{\frac{a^{3}+8}{a^{4}+3a^{3}+2a^{2}}}$$

Factor the expressions that are not already factored in \frac{a^{3}+8}{a^{4}+3a^{3}+2a^{2}}.

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

Cancel out a+2 in both numerator and denominator.

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

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

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

Cancel out a+1 in both numerator and denominator.

$$\frac{-a^{2}}{a^{2}-2a+4}$$

Expand

$$-\frac{a^{2}}{a^{2}-2a+4}$$

Short Solution Steps

Factor a^{2}-1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-1\right)\left(a+1\right) and a-1 is \left(a-1\right)\left(a+1\right). Multiply \frac{a+1}{a-1} times \frac{a+1}{a+1}.

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

Since \frac{a+3}{\left(a-1\right)\left(a+1\right)} and \frac{\left(a+1\right)\left(a+1\right)}{\left(a-1\right)\left(a+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in a+3-\left(a+1\right)\left(a+1\right).

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

Combine like terms in a+3-a^{2}-a-a-1.

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

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

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

Extract the negative sign in 1-a.

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

Cancel out a-1 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{a+1}{a+1}.

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

Since \frac{-\left(a+2\right)}{a+1} and \frac{a+1}{a+1} have the same denominator, add them by adding their numerators.

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

Do the multiplications in -\left(a+2\right)+a+1.

$$\frac{\frac{-a-2+a+1}{a+1}}{\frac{a^{3}+8}{a^{4}+3a^{3}+2a^{2}}}$$

Combine like terms in -a-2+a+1.

$$\frac{\frac{-1}{a+1}}{\frac{a^{3}+8}{a^{4}+3a^{3}+2a^{2}}}$$

Factor the expressions that are not already factored in \frac{a^{3}+8}{a^{4}+3a^{3}+2a^{2}}.

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

Cancel out a+2 in both numerator and denominator.

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

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

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

Cancel out a+1 in both numerator and denominator.

$$\frac{-a^{2}}{a^{2}-2a+4}$$