Solve 2x^2+3x-2quad-1/2x-1-1/x+2

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

Evaluate

$$\frac{4x^{4}+12x^{3}+x^{2}-15x+3}{\left(2x-1\right)\left(x+2\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 2x^{2}+3x-2 times \frac{2x-1}{2x-1}.

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

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

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

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

$$\frac{4x^{3}-2x^{2}+6x^{2}-3x-4x+2-1}{2x-1}-\frac{1}{x+2}$$

Combine like terms in 4x^{3}-2x^{2}+6x^{2}-3x-4x+2-1.

$$\frac{4x^{3}+4x^{2}-7x+1}{2x-1}-\frac{1}{x+2}$$

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

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

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

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

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

$$\frac{4x^{4}+8x^{3}+4x^{3}+8x^{2}-7x^{2}-14x+x+2-2x+1}{\left(2x-1\right)\left(x+2\right)}$$

Combine like terms in 4x^{4}+8x^{3}+4x^{3}+8x^{2}-7x^{2}-14x+x+2-2x+1.

$$\frac{4x^{4}+12x^{3}+x^{2}-15x+3}{\left(2x-1\right)\left(x+2\right)}$$

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

$$\frac{4x^{4}+12x^{3}+x^{2}-15x+3}{2x^{2}+3x-2}$$

Differentiate w.r.t. x

$$\frac{16x^{5}+60x^{4}+40x^{3}-39x^{2}-16x+21}{4x^{4}+12x^{3}+x^{2}-12x+4}$$