Solve 3/(x-1)(x+2)-1/(x+3)(x-4)

$$\frac { 3 } { ( x - 1 ) ( x + 2 ) } - \frac { 1 } { ( x + 3 ) ( x - 4 ) }$$

Evaluate

$$\frac{2\left(x^{2}-2x-17\right)}{\left(x-4\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+2\right) and \left(x+3\right)\left(x-4\right) is \left(x-4\right)\left(x-1\right)\left(x+2\right)\left(x+3\right). Multiply \frac{3}{\left(x-1\right)\left(x+2\right)} times \frac{\left(x-4\right)\left(x+3\right)}{\left(x-4\right)\left(x+3\right)}. Multiply \frac{1}{\left(x+3\right)\left(x-4\right)} times \frac{\left(x-1\right)\left(x+2\right)}{\left(x-1\right)\left(x+2\right)}.

$$\frac{3\left(x-4\right)\left(x+3\right)}{\left(x-4\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)}-\frac{\left(x-1\right)\left(x+2\right)}{\left(x-4\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)}$$

Since \frac{3\left(x-4\right)\left(x+3\right)}{\left(x-4\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)} and \frac{\left(x-1\right)\left(x+2\right)}{\left(x-4\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\left(x-4\right)\left(x+3\right)-\left(x-1\right)\left(x+2\right)}{\left(x-4\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)}$$

Do the multiplications in 3\left(x-4\right)\left(x+3\right)-\left(x-1\right)\left(x+2\right).

$$\frac{3x^{2}+9x-12x-36-x^{2}-2x+x+2}{\left(x-4\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)}$$

Combine like terms in 3x^{2}+9x-12x-36-x^{2}-2x+x+2.

$$\frac{2x^{2}-4x-34}{\left(x-4\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)}$$

Expand \left(x-4\right)\left(x-1\right)\left(x+2\right)\left(x+3\right).

$$\frac{2x^{2}-4x-34}{x^{4}-15x^{2}-10x+24}$$

Expand

$$\frac{2\left(x^{2}-2x-17\right)}{\left(x-4\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+2\right) and \left(x+3\right)\left(x-4\right) is \left(x-4\right)\left(x-1\right)\left(x+2\right)\left(x+3\right). Multiply \frac{3}{\left(x-1\right)\left(x+2\right)} times \frac{\left(x-4\right)\left(x+3\right)}{\left(x-4\right)\left(x+3\right)}. Multiply \frac{1}{\left(x+3\right)\left(x-4\right)} times \frac{\left(x-1\right)\left(x+2\right)}{\left(x-1\right)\left(x+2\right)}.

$$\frac{3\left(x-4\right)\left(x+3\right)}{\left(x-4\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)}-\frac{\left(x-1\right)\left(x+2\right)}{\left(x-4\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)}$$

Since \frac{3\left(x-4\right)\left(x+3\right)}{\left(x-4\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)} and \frac{\left(x-1\right)\left(x+2\right)}{\left(x-4\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\left(x-4\right)\left(x+3\right)-\left(x-1\right)\left(x+2\right)}{\left(x-4\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)}$$

Do the multiplications in 3\left(x-4\right)\left(x+3\right)-\left(x-1\right)\left(x+2\right).

$$\frac{3x^{2}+9x-12x-36-x^{2}-2x+x+2}{\left(x-4\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)}$$

Combine like terms in 3x^{2}+9x-12x-36-x^{2}-2x+x+2.

$$\frac{2x^{2}-4x-34}{\left(x-4\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)}$$

Expand \left(x-4\right)\left(x-1\right)\left(x+2\right)\left(x+3\right).

$$\frac{2x^{2}-4x-34}{x^{4}-15x^{2}-10x+24}$$