Solve 1/a^2+8a+15+2/a^2+2a-3

$$\frac { 1 } { a ^ { 2 } + 8 a + 15 } + \frac { 2 } { a ^ { 2 } + 2 a - 3 }$$

Evaluate

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

Short Solution Steps

Factor a^{2}+8a+15. Factor a^{2}+2a-3.

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

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

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

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

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

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

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

Combine like terms in a-1+2a+10.

$$\frac{3a+9}{\left(a-1\right)\left(a+3\right)\left(a+5\right)}$$

Factor the expressions that are not already factored in \frac{3a+9}{\left(a-1\right)\left(a+3\right)\left(a+5\right)}.

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

Cancel out a+3 in both numerator and denominator.

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

Expand \left(a-1\right)\left(a+5\right).

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

Differentiate w.r.t. a

$$\frac{6\left(-a-2\right)}{\left(\left(a-1\right)\left(a+5\right)\right)^{2}}$$