Solve 4x/3x^2-3x-18-4/2x

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

Evaluate

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

Short Solution Steps

Factor 3x^{2}-3x-18.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3\left(x-3\right)\left(x+2\right) and 2x is 6x\left(x-3\right)\left(x+2\right). Multiply \frac{4x}{3\left(x-3\right)\left(x+2\right)} times \frac{2x}{2x}. Multiply \frac{4}{2x} times \frac{3\left(x-3\right)\left(x+2\right)}{3\left(x-3\right)\left(x+2\right)}.

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

Since \frac{4x\times 2x}{6x\left(x-3\right)\left(x+2\right)} and \frac{4\times 3\left(x-3\right)\left(x+2\right)}{6x\left(x-3\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

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

$$\frac{8x^{2}-12x^{2}-24x+36x+72}{6x\left(x-3\right)\left(x+2\right)}$$

Combine like terms in 8x^{2}-12x^{2}-24x+36x+72.

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

Factor the expressions that are not already factored in \frac{-4x^{2}+12x+72}{6x\left(x-3\right)\left(x+2\right)}.

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

Cancel out 2 in both numerator and denominator.

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

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

$$\frac{2\left(x-6\right)\left(-x-3\right)}{3x^{3}-3x^{2}-18x}$$

Use the distributive property to multiply 2 by x-6.

$$\frac{\left(2x-12\right)\left(-x-3\right)}{3x^{3}-3x^{2}-18x}$$

Use the distributive property to multiply 2x-12 by -x-3 and combine like terms.

$$\frac{-2x^{2}+6x+36}{3x^{3}-3x^{2}-18x}$$

Factor

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