Solve x+3/x^2-3x+2-x^2-5x+6/3x+9

$$\frac { x + 3 } { x ^ { 2 } - 3 x + 2 } - \frac { x ^ { 2 } - 5 x + 6 } { 3 x + 9 }$$

Evaluate

$$\frac{15+46x-20x^{2}+8x^{3}-x^{4}}{3\left(x-2\right)\left(x-1\right)\left(x+3\right)}$$

Short Solution Steps

Factor x^{2}-3x+2. Factor 3x+9.

$$\frac{x+3}{\left(x-2\right)\left(x-1\right)}-\frac{x^{2}-5x+6}{3\left(x+3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x-1\right) and 3\left(x+3\right) is 3\left(x-2\right)\left(x-1\right)\left(x+3\right). Multiply \frac{x+3}{\left(x-2\right)\left(x-1\right)} times \frac{3\left(x+3\right)}{3\left(x+3\right)}. Multiply \frac{x^{2}-5x+6}{3\left(x+3\right)} times \frac{\left(x-2\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)}.

$$\frac{\left(x+3\right)\times 3\left(x+3\right)}{3\left(x-2\right)\left(x-1\right)\left(x+3\right)}-\frac{\left(x^{2}-5x+6\right)\left(x-2\right)\left(x-1\right)}{3\left(x-2\right)\left(x-1\right)\left(x+3\right)}$$

Since \frac{\left(x+3\right)\times 3\left(x+3\right)}{3\left(x-2\right)\left(x-1\right)\left(x+3\right)} and \frac{\left(x^{2}-5x+6\right)\left(x-2\right)\left(x-1\right)}{3\left(x-2\right)\left(x-1\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x+3\right)\times 3\left(x+3\right)-\left(x^{2}-5x+6\right)\left(x-2\right)\left(x-1\right)}{3\left(x-2\right)\left(x-1\right)\left(x+3\right)}$$

Do the multiplications in \left(x+3\right)\times 3\left(x+3\right)-\left(x^{2}-5x+6\right)\left(x-2\right)\left(x-1\right).

$$\frac{3x^{2}+9x+9x+27-x^{4}+3x^{3}-2x^{2}+5x^{3}-15x^{2}+10x-6x^{2}+18x-12}{3\left(x-2\right)\left(x-1\right)\left(x+3\right)}$$

Combine like terms in 3x^{2}+9x+9x+27-x^{4}+3x^{3}-2x^{2}+5x^{3}-15x^{2}+10x-6x^{2}+18x-12.

$$\frac{-20x^{2}+46x+15-x^{4}+8x^{3}}{3\left(x-2\right)\left(x-1\right)\left(x+3\right)}$$

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

$$\frac{-20x^{2}+46x+15-x^{4}+8x^{3}}{3x^{3}-21x+18}$$

Expand

$$-\frac{x^{4}-8x^{3}+20x^{2}-46x-15}{3\left(x+3\right)\left(x^{2}-3x+2\right)}$$

Short Solution Steps

Factor x^{2}-3x+2. Factor 3x+9.

$$\frac{x+3}{\left(x-2\right)\left(x-1\right)}-\frac{x^{2}-5x+6}{3\left(x+3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x-1\right) and 3\left(x+3\right) is 3\left(x-2\right)\left(x-1\right)\left(x+3\right). Multiply \frac{x+3}{\left(x-2\right)\left(x-1\right)} times \frac{3\left(x+3\right)}{3\left(x+3\right)}. Multiply \frac{x^{2}-5x+6}{3\left(x+3\right)} times \frac{\left(x-2\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)}.

$$\frac{\left(x+3\right)\times 3\left(x+3\right)}{3\left(x-2\right)\left(x-1\right)\left(x+3\right)}-\frac{\left(x^{2}-5x+6\right)\left(x-2\right)\left(x-1\right)}{3\left(x-2\right)\left(x-1\right)\left(x+3\right)}$$

Since \frac{\left(x+3\right)\times 3\left(x+3\right)}{3\left(x-2\right)\left(x-1\right)\left(x+3\right)} and \frac{\left(x^{2}-5x+6\right)\left(x-2\right)\left(x-1\right)}{3\left(x-2\right)\left(x-1\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x+3\right)\times 3\left(x+3\right)-\left(x^{2}-5x+6\right)\left(x-2\right)\left(x-1\right)}{3\left(x-2\right)\left(x-1\right)\left(x+3\right)}$$

Do the multiplications in \left(x+3\right)\times 3\left(x+3\right)-\left(x^{2}-5x+6\right)\left(x-2\right)\left(x-1\right).

$$\frac{3x^{2}+9x+9x+27-x^{4}+3x^{3}-2x^{2}+5x^{3}-15x^{2}+10x-6x^{2}+18x-12}{3\left(x-2\right)\left(x-1\right)\left(x+3\right)}$$

Combine like terms in 3x^{2}+9x+9x+27-x^{4}+3x^{3}-2x^{2}+5x^{3}-15x^{2}+10x-6x^{2}+18x-12.

$$\frac{-20x^{2}+46x+15-x^{4}+8x^{3}}{3\left(x-2\right)\left(x-1\right)\left(x+3\right)}$$

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

$$\frac{-20x^{2}+46x+15-x^{4}+8x^{3}}{3x^{3}-21x+18}$$