Solve x/4x^2+11x+6-2/8x^2+2x-3

$$\frac{ x }{ 4 { x }^{ 2 } +11x+6 } - \frac{ 2 }{ 8 { x }^{ 2 } +2x-3 }$$

Evaluate

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

Short Solution Steps

Factor 4x^{2}+11x+6. Factor 8x^{2}+2x-3.

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

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

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

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

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

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

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

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

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

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

$$\frac{2x^{2}-3x-4}{8x^{3}+18x^{2}+x-6}$$

Factor the expressions that are not already factored.

$$\frac{2\left(x-\left(-\frac{1}{4}\sqrt{41}+\frac{3}{4}\right)\right)\left(x-\left(\frac{1}{4}\sqrt{41}+\frac{3}{4}\right)\right)}{2\left(2x-1\right)\left(\frac{1}{2}x+1\right)\left(4x+3\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{\left(x-\left(-\frac{1}{4}\sqrt{41}+\frac{3}{4}\right)\right)\left(x-\left(\frac{1}{4}\sqrt{41}+\frac{3}{4}\right)\right)}{\left(2x-1\right)\left(\frac{1}{2}x+1\right)\left(4x+3\right)}$$

Expand the expression.

$$\frac{x^{2}-\frac{3}{2}x-2}{4x^{3}+9x^{2}+\frac{1}{2}x-3}$$

Differentiate w.r.t. x

$$\frac{2\left(11+60x+76x^{2}+24x^{3}-8x^{4}\right)}{\left(\left(2x-1\right)\left(x+2\right)\left(4x+3\right)\right)^{2}}$$