Solve 4x/x+1+5/2x-3-4/x

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

Evaluate

$$\frac{8x^{3}-15x^{2}+9x+12}{x\left(2x-3\right)\left(x+1\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+1 and 2x-3 is \left(2x-3\right)\left(x+1\right). Multiply \frac{4x}{x+1} times \frac{2x-3}{2x-3}. Multiply \frac{5}{2x-3} times \frac{x+1}{x+1}.

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

Since \frac{4x\left(2x-3\right)}{\left(2x-3\right)\left(x+1\right)} and \frac{5\left(x+1\right)}{\left(2x-3\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 4x\left(2x-3\right)+5\left(x+1\right).

$$\frac{8x^{2}-12x+5x+5}{\left(2x-3\right)\left(x+1\right)}-\frac{4}{x}$$

Combine like terms in 8x^{2}-12x+5x+5.

$$\frac{8x^{2}-7x+5}{\left(2x-3\right)\left(x+1\right)}-\frac{4}{x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(2x-3\right)\left(x+1\right) and x is x\left(2x-3\right)\left(x+1\right). Multiply \frac{8x^{2}-7x+5}{\left(2x-3\right)\left(x+1\right)} times \frac{x}{x}. Multiply \frac{4}{x} times \frac{\left(2x-3\right)\left(x+1\right)}{\left(2x-3\right)\left(x+1\right)}.

$$\frac{\left(8x^{2}-7x+5\right)x}{x\left(2x-3\right)\left(x+1\right)}-\frac{4\left(2x-3\right)\left(x+1\right)}{x\left(2x-3\right)\left(x+1\right)}$$

Since \frac{\left(8x^{2}-7x+5\right)x}{x\left(2x-3\right)\left(x+1\right)} and \frac{4\left(2x-3\right)\left(x+1\right)}{x\left(2x-3\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(8x^{2}-7x+5\right)x-4\left(2x-3\right)\left(x+1\right)}{x\left(2x-3\right)\left(x+1\right)}$$

Do the multiplications in \left(8x^{2}-7x+5\right)x-4\left(2x-3\right)\left(x+1\right).

$$\frac{8x^{3}-7x^{2}+5x-8x^{2}-8x+12x+12}{x\left(2x-3\right)\left(x+1\right)}$$

Combine like terms in 8x^{3}-7x^{2}+5x-8x^{2}-8x+12x+12.

$$\frac{8x^{3}-15x^{2}+9x+12}{x\left(2x-3\right)\left(x+1\right)}$$

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

$$\frac{8x^{3}-15x^{2}+9x+12}{2x^{3}-x^{2}-3x}$$

Differentiate w.r.t. x

$$\frac{2\left(11x^{4}-42x^{3}-9x^{2}+12x+18\right)}{\left(x\left(2x-3\right)\left(x+1\right)\right)^{2}}$$