Solve a-1/a+2-a^2-4/a^2-2a+1div1/a^2-1

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

Evaluate

$$\frac{9+10a+3a^{2}-3a^{3}-a^{4}}{\left(a-1\right)\left(a+2\right)}$$

Short Solution Steps

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

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

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

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

Cancel out a-1 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+2 and a-1 is \left(a-1\right)\left(a+2\right). Multiply \frac{a-1}{a+2} times \frac{a-1}{a-1}. Multiply \frac{\left(a-2\right)\left(a+1\right)\left(a+2\right)}{a-1} times \frac{a+2}{a+2}.

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

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

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

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

$$\frac{a^{2}-a-a+1-a^{4}-5a^{3}-8a^{2}-4a+2a^{3}+10a^{2}+16a+8}{\left(a-1\right)\left(a+2\right)}$$

Combine like terms in a^{2}-a-a+1-a^{4}-5a^{3}-8a^{2}-4a+2a^{3}+10a^{2}+16a+8.

$$\frac{3a^{2}+10a+9-a^{4}-3a^{3}}{\left(a-1\right)\left(a+2\right)}$$

Expand \left(a-1\right)\left(a+2\right).

$$\frac{3a^{2}+10a+9-a^{4}-3a^{3}}{a^{2}+a-2}$$

Expand

$$\frac{9+10a+3a^{2}-3a^{3}-a^{4}}{\left(a-1\right)\left(a+2\right)}$$

Short Solution Steps

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

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

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

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

Cancel out a-1 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+2 and a-1 is \left(a-1\right)\left(a+2\right). Multiply \frac{a-1}{a+2} times \frac{a-1}{a-1}. Multiply \frac{\left(a-2\right)\left(a+1\right)\left(a+2\right)}{a-1} times \frac{a+2}{a+2}.

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

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

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

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

$$\frac{a^{2}-a-a+1-a^{4}-5a^{3}-8a^{2}-4a+2a^{3}+10a^{2}+16a+8}{\left(a-1\right)\left(a+2\right)}$$

Combine like terms in a^{2}-a-a+1-a^{4}-5a^{3}-8a^{2}-4a+2a^{3}+10a^{2}+16a+8.

$$\frac{3a^{2}+10a+9-a^{4}-3a^{3}}{\left(a-1\right)\left(a+2\right)}$$

Expand \left(a-1\right)\left(a+2\right).

$$\frac{3a^{2}+10a+9-a^{4}-3a^{3}}{a^{2}+a-2}$$