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

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

Evaluate

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

Short Solution Steps

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

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

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

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

Do the multiplications in 4\left(x-2\right)-2\times 3x.

$$\frac{4x-8-6x}{3x\left(x-2\right)}+\frac{1}{2x+1}$$

Combine like terms in 4x-8-6x.

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

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

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

Since \frac{\left(-2x-8\right)\left(2x+1\right)}{3x\left(x-2\right)\left(2x+1\right)} and \frac{3x\left(x-2\right)}{3x\left(x-2\right)\left(2x+1\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(-2x-8\right)\left(2x+1\right)+3x\left(x-2\right).

$$\frac{-4x^{2}-2x-16x-8+3x^{2}-6x}{3x\left(x-2\right)\left(2x+1\right)}$$

Combine like terms in -4x^{2}-2x-16x-8+3x^{2}-6x.

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

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

$$\frac{-x^{2}-24x-8}{6x^{3}-9x^{2}-6x}$$

Factor

$$-\frac{\left(x-\left(-2\sqrt{34}-12\right)\right)\left(x-\left(2\sqrt{34}-12\right)\right)}{3x\left(x-2\right)\left(2x+1\right)}$$