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

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

Cancel out a+1 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out a-2 in both numerator and denominator.

$$\frac{-a-2}{a-2}$$

Expand

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

Cancel out a+1 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out a-2 in both numerator and denominator.

$$\frac{-a-2}{a-2}$$