Solve 6/a^2-2a-35-2/a^2+9a+20

$$\frac { 6 } { a ^ { 2 } - 2 a - 35 } - \frac { 2 } { a ^ { 2 } + 9 a + 20 }$$

Evaluate

$$\frac{2\left(2a+19\right)}{\left(a-7\right)\left(a+4\right)\left(a+5\right)}$$

Short Solution Steps

Factor a^{2}-2a-35. Factor a^{2}+9a+20.

$$\frac{6}{\left(a-7\right)\left(a+5\right)}-\frac{2}{\left(a+4\right)\left(a+5\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-7\right)\left(a+5\right) and \left(a+4\right)\left(a+5\right) is \left(a-7\right)\left(a+4\right)\left(a+5\right). Multiply \frac{6}{\left(a-7\right)\left(a+5\right)} times \frac{a+4}{a+4}. Multiply \frac{2}{\left(a+4\right)\left(a+5\right)} times \frac{a-7}{a-7}.

$$\frac{6\left(a+4\right)}{\left(a-7\right)\left(a+4\right)\left(a+5\right)}-\frac{2\left(a-7\right)}{\left(a-7\right)\left(a+4\right)\left(a+5\right)}$$

Since \frac{6\left(a+4\right)}{\left(a-7\right)\left(a+4\right)\left(a+5\right)} and \frac{2\left(a-7\right)}{\left(a-7\right)\left(a+4\right)\left(a+5\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{6\left(a+4\right)-2\left(a-7\right)}{\left(a-7\right)\left(a+4\right)\left(a+5\right)}$$

Do the multiplications in 6\left(a+4\right)-2\left(a-7\right).

$$\frac{6a+24-2a+14}{\left(a-7\right)\left(a+4\right)\left(a+5\right)}$$

Combine like terms in 6a+24-2a+14.

$$\frac{4a+38}{\left(a-7\right)\left(a+4\right)\left(a+5\right)}$$

Expand \left(a-7\right)\left(a+4\right)\left(a+5\right).

$$\frac{4a+38}{a^{3}+2a^{2}-43a-140}$$

Factor

$$\frac{2\left(2a+19\right)}{\left(a-7\right)\left(a+4\right)\left(a+5\right)}$$