Solve 4/(r-7)(r+3)-r+8/r-7

$$\frac { 4 } { ( r - 7 ) ( r + 3 ) } - \frac { r + 8 } { r - 7 }$$

Evaluate

$$-\frac{r^{2}+11r+20}{\left(r-7\right)\left(r+3\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(r-7\right)\left(r+3\right) and r-7 is \left(r-7\right)\left(r+3\right). Multiply \frac{r+8}{r-7} times \frac{r+3}{r+3}.

$$\frac{4}{\left(r-7\right)\left(r+3\right)}-\frac{\left(r+8\right)\left(r+3\right)}{\left(r-7\right)\left(r+3\right)}$$

Since \frac{4}{\left(r-7\right)\left(r+3\right)} and \frac{\left(r+8\right)\left(r+3\right)}{\left(r-7\right)\left(r+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4-\left(r+8\right)\left(r+3\right)}{\left(r-7\right)\left(r+3\right)}$$

Do the multiplications in 4-\left(r+8\right)\left(r+3\right).

$$\frac{4-r^{2}-3r-8r-24}{\left(r-7\right)\left(r+3\right)}$$

Combine like terms in 4-r^{2}-3r-8r-24.

$$\frac{-20-r^{2}-11r}{\left(r-7\right)\left(r+3\right)}$$

Expand \left(r-7\right)\left(r+3\right).

$$\frac{-20-r^{2}-11r}{r^{2}-4r-21}$$

Expand

$$-\frac{r^{2}+11r+20}{\left(r-7\right)\left(r+3\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(r-7\right)\left(r+3\right) and r-7 is \left(r-7\right)\left(r+3\right). Multiply \frac{r+8}{r-7} times \frac{r+3}{r+3}.

$$\frac{4}{\left(r-7\right)\left(r+3\right)}-\frac{\left(r+8\right)\left(r+3\right)}{\left(r-7\right)\left(r+3\right)}$$

Since \frac{4}{\left(r-7\right)\left(r+3\right)} and \frac{\left(r+8\right)\left(r+3\right)}{\left(r-7\right)\left(r+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4-\left(r+8\right)\left(r+3\right)}{\left(r-7\right)\left(r+3\right)}$$

Do the multiplications in 4-\left(r+8\right)\left(r+3\right).

$$\frac{4-r^{2}-3r-8r-24}{\left(r-7\right)\left(r+3\right)}$$

Combine like terms in 4-r^{2}-3r-8r-24.

$$\frac{-20-r^{2}-11r}{\left(r-7\right)\left(r+3\right)}$$

Expand \left(r-7\right)\left(r+3\right).

$$\frac{-20-r^{2}-11r}{r^{2}-4r-21}$$