Solve (1/x-3+2/x+4)+(14/x+4-5/x+1)

$$( \frac { 1 } { x - 3 } + \frac { 2 } { x + 4 } ) + ( \frac { 14 } { x + 4 } - \frac { 5 } { x + 1 } )$$

Evaluate

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

Short Solution Steps

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

$$\frac{x+4}{\left(x-3\right)\left(x+4\right)}+\frac{2\left(x-3\right)}{\left(x-3\right)\left(x+4\right)}+\frac{14}{x+4}-\frac{5}{x+1}$$

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

$$\frac{x+4+2\left(x-3\right)}{\left(x-3\right)\left(x+4\right)}+\frac{14}{x+4}-\frac{5}{x+1}$$

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

$$\frac{x+4+2x-6}{\left(x-3\right)\left(x+4\right)}+\frac{14}{x+4}-\frac{5}{x+1}$$

Combine like terms in x+4+2x-6.

$$\frac{3x-2}{\left(x-3\right)\left(x+4\right)}+\frac{14}{x+4}-\frac{5}{x+1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+4\right) and x+4 is \left(x-3\right)\left(x+4\right). Multiply \frac{14}{x+4} times \frac{x-3}{x-3}.

$$\frac{3x-2}{\left(x-3\right)\left(x+4\right)}+\frac{14\left(x-3\right)}{\left(x-3\right)\left(x+4\right)}-\frac{5}{x+1}$$

Since \frac{3x-2}{\left(x-3\right)\left(x+4\right)} and \frac{14\left(x-3\right)}{\left(x-3\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{3x-2+14\left(x-3\right)}{\left(x-3\right)\left(x+4\right)}-\frac{5}{x+1}$$

Do the multiplications in 3x-2+14\left(x-3\right).

$$\frac{3x-2+14x-42}{\left(x-3\right)\left(x+4\right)}-\frac{5}{x+1}$$

Combine like terms in 3x-2+14x-42.

$$\frac{17x-44}{\left(x-3\right)\left(x+4\right)}-\frac{5}{x+1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+4\right) and x+1 is \left(x-3\right)\left(x+1\right)\left(x+4\right). Multiply \frac{17x-44}{\left(x-3\right)\left(x+4\right)} times \frac{x+1}{x+1}. Multiply \frac{5}{x+1} times \frac{\left(x-3\right)\left(x+4\right)}{\left(x-3\right)\left(x+4\right)}.

$$\frac{\left(17x-44\right)\left(x+1\right)}{\left(x-3\right)\left(x+1\right)\left(x+4\right)}-\frac{5\left(x-3\right)\left(x+4\right)}{\left(x-3\right)\left(x+1\right)\left(x+4\right)}$$

Since \frac{\left(17x-44\right)\left(x+1\right)}{\left(x-3\right)\left(x+1\right)\left(x+4\right)} and \frac{5\left(x-3\right)\left(x+4\right)}{\left(x-3\right)\left(x+1\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(17x-44\right)\left(x+1\right)-5\left(x-3\right)\left(x+4\right)}{\left(x-3\right)\left(x+1\right)\left(x+4\right)}$$

Do the multiplications in \left(17x-44\right)\left(x+1\right)-5\left(x-3\right)\left(x+4\right).

$$\frac{17x^{2}+17x-44x-44-5x^{2}-20x+15x+60}{\left(x-3\right)\left(x+1\right)\left(x+4\right)}$$

Combine like terms in 17x^{2}+17x-44x-44-5x^{2}-20x+15x+60.

$$\frac{12x^{2}-32x+16}{\left(x-3\right)\left(x+1\right)\left(x+4\right)}$$

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

$$\frac{12x^{2}-32x+16}{x^{3}+2x^{2}-11x-12}$$

Differentiate w.r.t. x

$$\frac{4\left(140-88x-29x^{2}+16x^{3}-3x^{4}\right)}{\left(\left(x-3\right)\left(x+1\right)\left(x+4\right)\right)^{2}}$$