Solve 4x-12/(3-x)^2+5x+3/3x-x^2-1/x=

$$\frac { 4 x - 12 } { ( 3 - x ) ^ { 2 } } + \frac { 5 x + 3 } { 3 x - x ^ { 2 } } - \frac { 1 } { x } =$$

Evaluate

$$\frac{2}{3-x}$$

Short Solution Steps

Factor 3x-x^{2}.

$$\frac{4x-12}{\left(3-x\right)^{2}}+\frac{5x+3}{x\left(-x+3\right)}-\frac{1}{x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(3-x\right)^{2} and x\left(-x+3\right) is x\left(-x+3\right)\left(-x+3\right)^{2}. Multiply \frac{4x-12}{\left(3-x\right)^{2}} times \frac{x\left(-x+3\right)}{x\left(-x+3\right)}. Multiply \frac{5x+3}{x\left(-x+3\right)} times \frac{\left(-x+3\right)^{2}}{\left(-x+3\right)^{2}}.

$$\frac{\left(4x-12\right)x\left(-x+3\right)}{x\left(-x+3\right)\left(-x+3\right)^{2}}+\frac{\left(5x+3\right)\left(-x+3\right)^{2}}{x\left(-x+3\right)\left(-x+3\right)^{2}}-\frac{1}{x}$$

Since \frac{\left(4x-12\right)x\left(-x+3\right)}{x\left(-x+3\right)\left(-x+3\right)^{2}} and \frac{\left(5x+3\right)\left(-x+3\right)^{2}}{x\left(-x+3\right)\left(-x+3\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(4x-12\right)x\left(-x+3\right)+\left(5x+3\right)\left(-x+3\right)^{2}}{x\left(-x+3\right)\left(-x+3\right)^{2}}-\frac{1}{x}$$

Do the multiplications in \left(4x-12\right)x\left(-x+3\right)+\left(5x+3\right)\left(-x+3\right)^{2}.

$$\frac{-4x^{3}+12x^{2}+12x^{2}-36x+5x^{3}-30x^{2}+45x+3x^{2}-18x+27}{x\left(-x+3\right)\left(-x+3\right)^{2}}-\frac{1}{x}$$

Combine like terms in -4x^{3}+12x^{2}+12x^{2}-36x+5x^{3}-30x^{2}+45x+3x^{2}-18x+27.

$$\frac{x^{3}-3x^{2}-9x+27}{x\left(-x+3\right)\left(-x+3\right)^{2}}-\frac{1}{x}$$

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

$$\frac{x^{3}-3x^{2}-9x+27}{x\left(-x+3\right)\left(-x+3\right)^{2}}-\frac{\left(-x+3\right)\left(-x+3\right)^{2}}{x\left(-x+3\right)\left(-x+3\right)^{2}}$$

Since \frac{x^{3}-3x^{2}-9x+27}{x\left(-x+3\right)\left(-x+3\right)^{2}} and \frac{\left(-x+3\right)\left(-x+3\right)^{2}}{x\left(-x+3\right)\left(-x+3\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{3}-3x^{2}-9x+27-\left(-x+3\right)\left(-x+3\right)^{2}}{x\left(-x+3\right)\left(-x+3\right)^{2}}$$

Do the multiplications in x^{3}-3x^{2}-9x+27-\left(-x+3\right)\left(-x+3\right)^{2}.

$$\frac{x^{3}-3x^{2}-9x+27+x^{3}-6x^{2}+9x-3x^{2}+18x-27}{x\left(-x+3\right)\left(-x+3\right)^{2}}$$

Combine like terms in x^{3}-3x^{2}-9x+27+x^{3}-6x^{2}+9x-3x^{2}+18x-27.

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

Factor the expressions that are not already factored in \frac{2x^{3}-12x^{2}+18x}{x\left(-x+3\right)\left(-x+3\right)^{2}}.

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

Cancel out x in both numerator and denominator.

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

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

$$\frac{2\left(x-3\right)^{2}}{-x^{3}+9x^{2}-27x+27}$$

Factor the expressions that are not already factored.

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

Cancel out \left(x-3\right)^{2} in both numerator and denominator.

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

Expand

$$-\frac{2}{x-3}$$

Short Solution Steps

Factor 3x-x^{2}.

$$\frac{4x-12}{\left(3-x\right)^{2}}+\frac{5x+3}{x\left(-x+3\right)}-\frac{1}{x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(3-x\right)^{2} and x\left(-x+3\right) is x\left(-x+3\right)\left(-x+3\right)^{2}. Multiply \frac{4x-12}{\left(3-x\right)^{2}} times \frac{x\left(-x+3\right)}{x\left(-x+3\right)}. Multiply \frac{5x+3}{x\left(-x+3\right)} times \frac{\left(-x+3\right)^{2}}{\left(-x+3\right)^{2}}.

$$\frac{\left(4x-12\right)x\left(-x+3\right)}{x\left(-x+3\right)\left(-x+3\right)^{2}}+\frac{\left(5x+3\right)\left(-x+3\right)^{2}}{x\left(-x+3\right)\left(-x+3\right)^{2}}-\frac{1}{x}$$

Since \frac{\left(4x-12\right)x\left(-x+3\right)}{x\left(-x+3\right)\left(-x+3\right)^{2}} and \frac{\left(5x+3\right)\left(-x+3\right)^{2}}{x\left(-x+3\right)\left(-x+3\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(4x-12\right)x\left(-x+3\right)+\left(5x+3\right)\left(-x+3\right)^{2}}{x\left(-x+3\right)\left(-x+3\right)^{2}}-\frac{1}{x}$$

Do the multiplications in \left(4x-12\right)x\left(-x+3\right)+\left(5x+3\right)\left(-x+3\right)^{2}.

$$\frac{-4x^{3}+12x^{2}+12x^{2}-36x+5x^{3}-30x^{2}+45x+3x^{2}-18x+27}{x\left(-x+3\right)\left(-x+3\right)^{2}}-\frac{1}{x}$$

Combine like terms in -4x^{3}+12x^{2}+12x^{2}-36x+5x^{3}-30x^{2}+45x+3x^{2}-18x+27.

$$\frac{x^{3}-3x^{2}-9x+27}{x\left(-x+3\right)\left(-x+3\right)^{2}}-\frac{1}{x}$$

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

$$\frac{x^{3}-3x^{2}-9x+27}{x\left(-x+3\right)\left(-x+3\right)^{2}}-\frac{\left(-x+3\right)\left(-x+3\right)^{2}}{x\left(-x+3\right)\left(-x+3\right)^{2}}$$

Since \frac{x^{3}-3x^{2}-9x+27}{x\left(-x+3\right)\left(-x+3\right)^{2}} and \frac{\left(-x+3\right)\left(-x+3\right)^{2}}{x\left(-x+3\right)\left(-x+3\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{3}-3x^{2}-9x+27-\left(-x+3\right)\left(-x+3\right)^{2}}{x\left(-x+3\right)\left(-x+3\right)^{2}}$$

Do the multiplications in x^{3}-3x^{2}-9x+27-\left(-x+3\right)\left(-x+3\right)^{2}.

$$\frac{x^{3}-3x^{2}-9x+27+x^{3}-6x^{2}+9x-3x^{2}+18x-27}{x\left(-x+3\right)\left(-x+3\right)^{2}}$$

Combine like terms in x^{3}-3x^{2}-9x+27+x^{3}-6x^{2}+9x-3x^{2}+18x-27.

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

Factor the expressions that are not already factored in \frac{2x^{3}-12x^{2}+18x}{x\left(-x+3\right)\left(-x+3\right)^{2}}.

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

Cancel out x in both numerator and denominator.

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

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

$$\frac{2\left(x-3\right)^{2}}{-x^{3}+9x^{2}-27x+27}$$

Factor the expressions that are not already factored.

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

Cancel out \left(x-3\right)^{2} in both numerator and denominator.

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