Solve 3x^2-x^2/2x^2-x-6-12

$$3 { x }^{ 2 } - \frac{ { x }^{ 2 } }{ 2 { x }^{ 2 } -x-6 } -12$$

Evaluate

$$\frac{6x^{4}-3x^{3}-43x^{2}+12x+72}{\left(x-2\right)\left(2x+3\right)}$$

Short Solution Steps

Factor 2x^{2}-x-6.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 3x^{2}-12 times \frac{\left(x-2\right)\left(2x+3\right)}{\left(x-2\right)\left(2x+3\right)}.

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

Since \frac{\left(3x^{2}-12\right)\left(x-2\right)\left(2x+3\right)}{\left(x-2\right)\left(2x+3\right)} and \frac{x^{2}}{\left(x-2\right)\left(2x+3\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(3x^{2}-12\right)\left(x-2\right)\left(2x+3\right)-x^{2}.

$$\frac{6x^{4}-3x^{3}-18x^{2}-24x^{2}+12x+72-x^{2}}{\left(x-2\right)\left(2x+3\right)}$$

Combine like terms in 6x^{4}-3x^{3}-18x^{2}-24x^{2}+12x+72-x^{2}.

$$\frac{6x^{4}-3x^{3}-43x^{2}+12x+72}{\left(x-2\right)\left(2x+3\right)}$$

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

$$\frac{6x^{4}-3x^{3}-43x^{2}+12x+72}{2x^{2}-x-6}$$

Differentiate w.r.t. x

$$\frac{x\left(24x^{4}-24x^{3}-138x^{2}+73x+228\right)}{\left(\left(x-2\right)\left(2x+3\right)\right)^{2}}$$