Solve f-1/f+4-f-1/f+1

$$\frac { f - 1 } { f + 4 } - \frac { f - 1 } { f + 1 }$$

Evaluate

$$-\frac{3\left(f-1\right)}{\left(f+1\right)\left(f+4\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of f+4 and f+1 is \left(f+1\right)\left(f+4\right). Multiply \frac{f-1}{f+4} times \frac{f+1}{f+1}. Multiply \frac{f-1}{f+1} times \frac{f+4}{f+4}.

$$\frac{\left(f-1\right)\left(f+1\right)}{\left(f+1\right)\left(f+4\right)}-\frac{\left(f-1\right)\left(f+4\right)}{\left(f+1\right)\left(f+4\right)}$$

Since \frac{\left(f-1\right)\left(f+1\right)}{\left(f+1\right)\left(f+4\right)} and \frac{\left(f-1\right)\left(f+4\right)}{\left(f+1\right)\left(f+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(f-1\right)\left(f+1\right)-\left(f-1\right)\left(f+4\right)}{\left(f+1\right)\left(f+4\right)}$$

Do the multiplications in \left(f-1\right)\left(f+1\right)-\left(f-1\right)\left(f+4\right).

$$\frac{f^{2}+f-f-1-f^{2}-4f+f+4}{\left(f+1\right)\left(f+4\right)}$$

Combine like terms in f^{2}+f-f-1-f^{2}-4f+f+4.

$$\frac{-3f+3}{\left(f+1\right)\left(f+4\right)}$$

Expand \left(f+1\right)\left(f+4\right).

$$\frac{-3f+3}{f^{2}+5f+4}$$

Expand

$$-\frac{3\left(f-1\right)}{\left(f+1\right)\left(f+4\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of f+4 and f+1 is \left(f+1\right)\left(f+4\right). Multiply \frac{f-1}{f+4} times \frac{f+1}{f+1}. Multiply \frac{f-1}{f+1} times \frac{f+4}{f+4}.

$$\frac{\left(f-1\right)\left(f+1\right)}{\left(f+1\right)\left(f+4\right)}-\frac{\left(f-1\right)\left(f+4\right)}{\left(f+1\right)\left(f+4\right)}$$

Since \frac{\left(f-1\right)\left(f+1\right)}{\left(f+1\right)\left(f+4\right)} and \frac{\left(f-1\right)\left(f+4\right)}{\left(f+1\right)\left(f+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(f-1\right)\left(f+1\right)-\left(f-1\right)\left(f+4\right)}{\left(f+1\right)\left(f+4\right)}$$

Do the multiplications in \left(f-1\right)\left(f+1\right)-\left(f-1\right)\left(f+4\right).

$$\frac{f^{2}+f-f-1-f^{2}-4f+f+4}{\left(f+1\right)\left(f+4\right)}$$

Combine like terms in f^{2}+f-f-1-f^{2}-4f+f+4.

$$\frac{-3f+3}{\left(f+1\right)\left(f+4\right)}$$

Expand \left(f+1\right)\left(f+4\right).

$$\frac{-3f+3}{f^{2}+5f+4}$$