Solve 3/a+1-a+1/frac{a^2-4a+4{a+1}}

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

Evaluate

$$\frac{13-14a+4a^{2}+3a^{3}-a^{4}}{\left(a+1\right)\left(a-2\right)^{2}}$$

Short Solution Steps

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

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

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

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

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

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

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

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

Factor a^{2}-4a+4.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+1 and \left(a-2\right)^{2} is \left(a+1\right)\left(a-2\right)^{2}. Multiply \frac{3-a^{2}-a}{a+1} times \frac{\left(a-2\right)^{2}}{\left(a-2\right)^{2}}. Multiply \frac{a+1}{\left(a-2\right)^{2}} times \frac{a+1}{a+1}.

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

Since \frac{\left(3-a^{2}-a\right)\left(a-2\right)^{2}}{\left(a+1\right)\left(a-2\right)^{2}} and \frac{\left(a+1\right)\left(a+1\right)}{\left(a+1\right)\left(a-2\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(3-a^{2}-a\right)\left(a-2\right)^{2}+\left(a+1\right)\left(a+1\right).

$$\frac{3a^{2}-12a+12-a^{4}+4a^{3}-4a^{2}-a^{3}+4a^{2}-4a+a^{2}+a+a+1}{\left(a+1\right)\left(a-2\right)^{2}}$$

Combine like terms in 3a^{2}-12a+12-a^{4}+4a^{3}-4a^{2}-a^{3}+4a^{2}-4a+a^{2}+a+a+1.

$$\frac{4a^{2}-14a+13-a^{4}+3a^{3}}{\left(a+1\right)\left(a-2\right)^{2}}$$

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

$$\frac{4a^{2}-14a+13-a^{4}+3a^{3}}{a^{3}-3a^{2}+4}$$

Expand

$$\frac{13-14a+4a^{2}+3a^{3}-a^{4}}{\left(a+1\right)\left(a^{2}-4a+4\right)}$$

Short Solution Steps

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

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

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

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

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

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

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

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

Factor a^{2}-4a+4.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+1 and \left(a-2\right)^{2} is \left(a+1\right)\left(a-2\right)^{2}. Multiply \frac{3-a^{2}-a}{a+1} times \frac{\left(a-2\right)^{2}}{\left(a-2\right)^{2}}. Multiply \frac{a+1}{\left(a-2\right)^{2}} times \frac{a+1}{a+1}.

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

Since \frac{\left(3-a^{2}-a\right)\left(a-2\right)^{2}}{\left(a+1\right)\left(a-2\right)^{2}} and \frac{\left(a+1\right)\left(a+1\right)}{\left(a+1\right)\left(a-2\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(3-a^{2}-a\right)\left(a-2\right)^{2}+\left(a+1\right)\left(a+1\right).

$$\frac{3a^{2}-12a+12-a^{4}+4a^{3}-4a^{2}-a^{3}+4a^{2}-4a+a^{2}+a+a+1}{\left(a+1\right)\left(a-2\right)^{2}}$$

Combine like terms in 3a^{2}-12a+12-a^{4}+4a^{3}-4a^{2}-a^{3}+4a^{2}-4a+a^{2}+a+a+1.

$$\frac{4a^{2}-14a+13-a^{4}+3a^{3}}{\left(a+1\right)\left(a-2\right)^{2}}$$

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

$$\frac{4a^{2}-14a+13-a^{4}+3a^{3}}{a^{3}-3a^{2}+4}$$