Solve 4w/w^2+9w+14-2w+5/w^2+3w-28

$$\frac { 4 w } { w ^ { 2 } + 9 w + 14 } - \frac { 2 w + 5 } { w ^ { 2 } + 3 w - 28 }$$

Evaluate

$$\frac{2w^{2}-25w-10}{\left(w-4\right)\left(w+2\right)\left(w+7\right)}$$

Short Solution Steps

Factor w^{2}+9w+14. Factor w^{2}+3w-28.

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

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

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

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

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

Do the multiplications in 4w\left(w-4\right)-\left(2w+5\right)\left(w+2\right).

$$\frac{4w^{2}-16w-2w^{2}-4w-5w-10}{\left(w-4\right)\left(w+2\right)\left(w+7\right)}$$

Combine like terms in 4w^{2}-16w-2w^{2}-4w-5w-10.

$$\frac{2w^{2}-25w-10}{\left(w-4\right)\left(w+2\right)\left(w+7\right)}$$

Expand \left(w-4\right)\left(w+2\right)\left(w+7\right).

$$\frac{2w^{2}-25w-10}{w^{3}+5w^{2}-22w-56}$$

Expand

$$\frac{2w^{2}-25w-10}{\left(w-4\right)\left(w+2\right)\left(w+7\right)}$$

Short Solution Steps

Factor w^{2}+9w+14. Factor w^{2}+3w-28.

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

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

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

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

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

Do the multiplications in 4w\left(w-4\right)-\left(2w+5\right)\left(w+2\right).

$$\frac{4w^{2}-16w-2w^{2}-4w-5w-10}{\left(w-4\right)\left(w+2\right)\left(w+7\right)}$$

Combine like terms in 4w^{2}-16w-2w^{2}-4w-5w-10.

$$\frac{2w^{2}-25w-10}{\left(w-4\right)\left(w+2\right)\left(w+7\right)}$$

Expand \left(w-4\right)\left(w+2\right)\left(w+7\right).

$$\frac{2w^{2}-25w-10}{w^{3}+5w^{2}-22w-56}$$