Solve p+3/p-q+1/p+2

$$\frac { p + 3 } { p - q } + \frac { 1 } { p + 2 }$$

Evaluate

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

Short Solution Steps

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

$$\frac{\left(p+3\right)\left(p+2\right)}{\left(p+2\right)\left(p-q\right)}+\frac{p-q}{\left(p+2\right)\left(p-q\right)}$$

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

$$\frac{\left(p+3\right)\left(p+2\right)+p-q}{\left(p+2\right)\left(p-q\right)}$$

Do the multiplications in \left(p+3\right)\left(p+2\right)+p-q.

$$\frac{p^{2}+2p+3p+6+p-q}{\left(p+2\right)\left(p-q\right)}$$

Combine like terms in p^{2}+2p+3p+6+p-q.

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

Expand \left(p+2\right)\left(p-q\right).

$$\frac{p^{2}+6p+6-q}{p^{2}-pq+2p-2q}$$

Expand

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

Short Solution Steps

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

$$\frac{\left(p+3\right)\left(p+2\right)}{\left(p+2\right)\left(p-q\right)}+\frac{p-q}{\left(p+2\right)\left(p-q\right)}$$

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

$$\frac{\left(p+3\right)\left(p+2\right)+p-q}{\left(p+2\right)\left(p-q\right)}$$

Do the multiplications in \left(p+3\right)\left(p+2\right)+p-q.

$$\frac{p^{2}+2p+3p+6+p-q}{\left(p+2\right)\left(p-q\right)}$$

Combine like terms in p^{2}+2p+3p+6+p-q.

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

Expand \left(p+2\right)\left(p-q\right).

$$\frac{p^{2}+6p+6-q}{p^{2}-pq+2p-2q}$$