Solve 9/w^2-w-12+w/w^2+8w+15

$$\frac { 9 } { w ^ { 2 } - w - 12 } + \frac { w } { w ^ { 2 } + 8 w + 15 }$$

Evaluate

$$\frac{w^{2}+5w+45}{\left(w-4\right)\left(w+3\right)\left(w+5\right)}$$

Short Solution Steps

Factor w^{2}-w-12. Factor w^{2}+8w+15.

$$\frac{9}{\left(w-4\right)\left(w+3\right)}+\frac{w}{\left(w+3\right)\left(w+5\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(w-4\right)\left(w+3\right) and \left(w+3\right)\left(w+5\right) is \left(w-4\right)\left(w+3\right)\left(w+5\right). Multiply \frac{9}{\left(w-4\right)\left(w+3\right)} times \frac{w+5}{w+5}. Multiply \frac{w}{\left(w+3\right)\left(w+5\right)} times \frac{w-4}{w-4}.

$$\frac{9\left(w+5\right)}{\left(w-4\right)\left(w+3\right)\left(w+5\right)}+\frac{w\left(w-4\right)}{\left(w-4\right)\left(w+3\right)\left(w+5\right)}$$

Since \frac{9\left(w+5\right)}{\left(w-4\right)\left(w+3\right)\left(w+5\right)} and \frac{w\left(w-4\right)}{\left(w-4\right)\left(w+3\right)\left(w+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{9\left(w+5\right)+w\left(w-4\right)}{\left(w-4\right)\left(w+3\right)\left(w+5\right)}$$

Do the multiplications in 9\left(w+5\right)+w\left(w-4\right).

$$\frac{9w+45+w^{2}-4w}{\left(w-4\right)\left(w+3\right)\left(w+5\right)}$$

Combine like terms in 9w+45+w^{2}-4w.

$$\frac{5w+45+w^{2}}{\left(w-4\right)\left(w+3\right)\left(w+5\right)}$$

Expand \left(w-4\right)\left(w+3\right)\left(w+5\right).

$$\frac{5w+45+w^{2}}{w^{3}+4w^{2}-17w-60}$$

Differentiate w.r.t. w

$$\frac{465-480w-172w^{2}-10w^{3}-w^{4}}{\left(\left(w-4\right)\left(w+3\right)\left(w+5\right)\right)^{2}}$$