Solve 1/a-5+a/a^2-4a-5+a+3/a^2+2a+1

$$\frac { 1 } { a - 5 } + \frac { a } { a ^ { 2 } - 4 a - 5 } + \frac { a + 3 } { a ^ { 2 } + 2 a + 1 }$$

Evaluate

$$\frac{\left(a-2\right)\left(3a+7\right)}{\left(a-5\right)\left(a+1\right)^{2}}$$

Short Solution Steps

Factor a^{2}-4a-5.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-5 and \left(a-5\right)\left(a+1\right) is \left(a-5\right)\left(a+1\right). Multiply \frac{1}{a-5} times \frac{a+1}{a+1}.

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

Since \frac{a+1}{\left(a-5\right)\left(a+1\right)} and \frac{a}{\left(a-5\right)\left(a+1\right)} have the same denominator, add them by adding their numerators.

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

Combine like terms in a+1+a.

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

Factor a^{2}+2a+1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-5\right)\left(a+1\right) and \left(a+1\right)^{2} is \left(a-5\right)\left(a+1\right)^{2}. Multiply \frac{2a+1}{\left(a-5\right)\left(a+1\right)} times \frac{a+1}{a+1}. Multiply \frac{a+3}{\left(a+1\right)^{2}} times \frac{a-5}{a-5}.

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

Since \frac{\left(2a+1\right)\left(a+1\right)}{\left(a-5\right)\left(a+1\right)^{2}} and \frac{\left(a+3\right)\left(a-5\right)}{\left(a-5\right)\left(a+1\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(2a+1\right)\left(a+1\right)+\left(a+3\right)\left(a-5\right).

$$\frac{2a^{2}+2a+a+1+a^{2}-5a+3a-15}{\left(a-5\right)\left(a+1\right)^{2}}$$

Combine like terms in 2a^{2}+2a+a+1+a^{2}-5a+3a-15.

$$\frac{3a^{2}+a-14}{\left(a-5\right)\left(a+1\right)^{2}}$$

Expand \left(a-5\right)\left(a+1\right)^{2}.

$$\frac{3a^{2}+a-14}{a^{3}-3a^{2}-9a-5}$$

Expand

$$\frac{3a^{2}+a-14}{\left(a-5\right)\left(a+1\right)^{2}}$$

Short Solution Steps

Factor a^{2}-4a-5.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-5 and \left(a-5\right)\left(a+1\right) is \left(a-5\right)\left(a+1\right). Multiply \frac{1}{a-5} times \frac{a+1}{a+1}.

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

Since \frac{a+1}{\left(a-5\right)\left(a+1\right)} and \frac{a}{\left(a-5\right)\left(a+1\right)} have the same denominator, add them by adding their numerators.

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

Combine like terms in a+1+a.

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

Factor a^{2}+2a+1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-5\right)\left(a+1\right) and \left(a+1\right)^{2} is \left(a-5\right)\left(a+1\right)^{2}. Multiply \frac{2a+1}{\left(a-5\right)\left(a+1\right)} times \frac{a+1}{a+1}. Multiply \frac{a+3}{\left(a+1\right)^{2}} times \frac{a-5}{a-5}.

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

Since \frac{\left(2a+1\right)\left(a+1\right)}{\left(a-5\right)\left(a+1\right)^{2}} and \frac{\left(a+3\right)\left(a-5\right)}{\left(a-5\right)\left(a+1\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(2a+1\right)\left(a+1\right)+\left(a+3\right)\left(a-5\right).

$$\frac{2a^{2}+2a+a+1+a^{2}-5a+3a-15}{\left(a-5\right)\left(a+1\right)^{2}}$$

Combine like terms in 2a^{2}+2a+a+1+a^{2}-5a+3a-15.

$$\frac{3a^{2}+a-14}{\left(a-5\right)\left(a+1\right)^{2}}$$

Expand \left(a-5\right)\left(a+1\right)^{2}.

$$\frac{3a^{2}+a-14}{a^{3}-3a^{2}-9a-5}$$