Solve x/x+4+3-3x/x-1

$$\frac{ x }{ x+4 } +3- \frac{ 3x }{ x-1 }$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{x+4}{x+4}.

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

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

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

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

$$\frac{x+3x+12}{x+4}-\frac{3x}{x-1}$$

Combine like terms in x+3x+12.

$$\frac{4x+12}{x+4}-\frac{3x}{x-1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+4 and x-1 is \left(x-1\right)\left(x+4\right). Multiply \frac{4x+12}{x+4} times \frac{x-1}{x-1}. Multiply \frac{3x}{x-1} times \frac{x+4}{x+4}.

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

Since \frac{\left(4x+12\right)\left(x-1\right)}{\left(x-1\right)\left(x+4\right)} and \frac{3x\left(x+4\right)}{\left(x-1\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(4x+12\right)\left(x-1\right)-3x\left(x+4\right).

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

Combine like terms in 4x^{2}-4x+12x-12-3x^{2}-12x.

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

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

$$\frac{x^{2}-4x-12}{x^{2}+3x-4}$$

Differentiate w.r.t. x

$$\frac{7x^{2}+16x+52}{\left(\left(x-1\right)\left(x+4\right)\right)^{2}}$$