Solve 6-p/16-p^2+p+2/p-4

$$\frac { 6 - p } { 16 - p ^ { 2 } } + \frac { p + 2 } { p - 4 }$$

Evaluate

$$\frac{p^{2}+7p+2}{p^{2}-16}$$

Short Solution Steps

Factor 16-p^{2}.

$$\frac{6-p}{\left(p-4\right)\left(-p-4\right)}+\frac{p+2}{p-4}$$

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

$$\frac{6-p}{\left(p-4\right)\left(-p-4\right)}+\frac{\left(p+2\right)\left(-p-4\right)}{\left(p-4\right)\left(-p-4\right)}$$

Since \frac{6-p}{\left(p-4\right)\left(-p-4\right)} and \frac{\left(p+2\right)\left(-p-4\right)}{\left(p-4\right)\left(-p-4\right)} have the same denominator, add them by adding their numerators.

$$\frac{6-p+\left(p+2\right)\left(-p-4\right)}{\left(p-4\right)\left(-p-4\right)}$$

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

$$\frac{6-p-p^{2}-4p-2p-8}{\left(p-4\right)\left(-p-4\right)}$$

Combine like terms in 6-p-p^{2}-4p-2p-8.

$$\frac{-2-7p-p^{2}}{\left(p-4\right)\left(-p-4\right)}$$

Expand \left(p-4\right)\left(-p-4\right).

$$\frac{-2-7p-p^{2}}{-p^{2}+16}$$

Expand

$$\frac{p^{2}+7p+2}{p^{2}-16}$$

Short Solution Steps

Factor 16-p^{2}.

$$\frac{6-p}{\left(p-4\right)\left(-p-4\right)}+\frac{p+2}{p-4}$$

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

$$\frac{6-p}{\left(p-4\right)\left(-p-4\right)}+\frac{\left(p+2\right)\left(-p-4\right)}{\left(p-4\right)\left(-p-4\right)}$$

Since \frac{6-p}{\left(p-4\right)\left(-p-4\right)} and \frac{\left(p+2\right)\left(-p-4\right)}{\left(p-4\right)\left(-p-4\right)} have the same denominator, add them by adding their numerators.

$$\frac{6-p+\left(p+2\right)\left(-p-4\right)}{\left(p-4\right)\left(-p-4\right)}$$

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

$$\frac{6-p-p^{2}-4p-2p-8}{\left(p-4\right)\left(-p-4\right)}$$

Combine like terms in 6-p-p^{2}-4p-2p-8.

$$\frac{-2-7p-p^{2}}{\left(p-4\right)\left(-p-4\right)}$$

Expand \left(p-4\right)\left(-p-4\right).

$$\frac{-2-7p-p^{2}}{-p^{2}+16}$$