Solve 12x^2-12x+4/(3x-2)(3x+2)(x-1)-(x+7)(x-1)/3x-2+2-x/(3x+2)(x-1)

$$\frac { 12 x ^ { 2 } - 12 x + 4 } { ( 3 x - 2 ) ( 3 x + 2 ) ( x - 1 ) } - \frac { ( x + 7 ) ( x - 1 ) } { 3 x - 2 } + \frac { 2 - x } { ( 3 x + 2 ) ( x - 1 ) }$$

Evaluate

$$\frac{-3x^{4}-17x^{3}+38x^{2}+x-14}{\left(x-1\right)\left(9x^{2}-4\right)}$$

Short Solution Steps

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

$$\frac{12x^{2}-12x+4}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)}-\frac{\left(x+7\right)\left(x-1\right)\left(x-1\right)\left(3x+2\right)}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)}+\frac{2-x}{\left(3x+2\right)\left(x-1\right)}$$

Since \frac{12x^{2}-12x+4}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)} and \frac{\left(x+7\right)\left(x-1\right)\left(x-1\right)\left(3x+2\right)}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{12x^{2}-12x+4-\left(x+7\right)\left(x-1\right)\left(x-1\right)\left(3x+2\right)}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)}+\frac{2-x}{\left(3x+2\right)\left(x-1\right)}$$

Do the multiplications in 12x^{2}-12x+4-\left(x+7\right)\left(x-1\right)\left(x-1\right)\left(3x+2\right).

$$\frac{12x^{2}-12x+4-3x^{4}+4x^{3}+x^{2}-2x-21x^{3}+28x^{2}+7x-14}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)}+\frac{2-x}{\left(3x+2\right)\left(x-1\right)}$$

Combine like terms in 12x^{2}-12x+4-3x^{4}+4x^{3}+x^{2}-2x-21x^{3}+28x^{2}+7x-14.

$$\frac{41x^{2}-7x-10-3x^{4}-17x^{3}}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)}+\frac{2-x}{\left(3x+2\right)\left(x-1\right)}$$

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

$$\frac{41x^{2}-7x-10-3x^{4}-17x^{3}}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)}+\frac{\left(2-x\right)\left(3x-2\right)}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)}$$

Since \frac{41x^{2}-7x-10-3x^{4}-17x^{3}}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)} and \frac{\left(2-x\right)\left(3x-2\right)}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{41x^{2}-7x-10-3x^{4}-17x^{3}+\left(2-x\right)\left(3x-2\right)}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)}$$

Do the multiplications in 41x^{2}-7x-10-3x^{4}-17x^{3}+\left(2-x\right)\left(3x-2\right).

$$\frac{41x^{2}-7x-10-3x^{4}-17x^{3}+6x-4-3x^{2}+2x}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)}$$

Combine like terms in 41x^{2}-7x-10-3x^{4}-17x^{3}+6x-4-3x^{2}+2x.

$$\frac{38x^{2}+x-14-3x^{4}-17x^{3}}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)}$$

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

$$\frac{38x^{2}+x-14-3x^{4}-17x^{3}}{9x^{3}-9x^{2}-4x+4}$$

Expand

$$\frac{-3x^{4}-17x^{3}+38x^{2}+x-14}{\left(x-1\right)\left(9x^{2}-4\right)}$$

Short Solution Steps

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

$$\frac{12x^{2}-12x+4}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)}-\frac{\left(x+7\right)\left(x-1\right)\left(x-1\right)\left(3x+2\right)}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)}+\frac{2-x}{\left(3x+2\right)\left(x-1\right)}$$

Since \frac{12x^{2}-12x+4}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)} and \frac{\left(x+7\right)\left(x-1\right)\left(x-1\right)\left(3x+2\right)}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{12x^{2}-12x+4-\left(x+7\right)\left(x-1\right)\left(x-1\right)\left(3x+2\right)}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)}+\frac{2-x}{\left(3x+2\right)\left(x-1\right)}$$

Do the multiplications in 12x^{2}-12x+4-\left(x+7\right)\left(x-1\right)\left(x-1\right)\left(3x+2\right).

$$\frac{12x^{2}-12x+4-3x^{4}+4x^{3}+x^{2}-2x-21x^{3}+28x^{2}+7x-14}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)}+\frac{2-x}{\left(3x+2\right)\left(x-1\right)}$$

Combine like terms in 12x^{2}-12x+4-3x^{4}+4x^{3}+x^{2}-2x-21x^{3}+28x^{2}+7x-14.

$$\frac{41x^{2}-7x-10-3x^{4}-17x^{3}}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)}+\frac{2-x}{\left(3x+2\right)\left(x-1\right)}$$

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

$$\frac{41x^{2}-7x-10-3x^{4}-17x^{3}}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)}+\frac{\left(2-x\right)\left(3x-2\right)}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)}$$

Since \frac{41x^{2}-7x-10-3x^{4}-17x^{3}}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)} and \frac{\left(2-x\right)\left(3x-2\right)}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{41x^{2}-7x-10-3x^{4}-17x^{3}+\left(2-x\right)\left(3x-2\right)}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)}$$

Do the multiplications in 41x^{2}-7x-10-3x^{4}-17x^{3}+\left(2-x\right)\left(3x-2\right).

$$\frac{41x^{2}-7x-10-3x^{4}-17x^{3}+6x-4-3x^{2}+2x}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)}$$

Combine like terms in 41x^{2}-7x-10-3x^{4}-17x^{3}+6x-4-3x^{2}+2x.

$$\frac{38x^{2}+x-14-3x^{4}-17x^{3}}{\left(x-1\right)\left(3x-2\right)\left(3x+2\right)}$$

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

$$\frac{38x^{2}+x-14-3x^{4}-17x^{3}}{9x^{3}-9x^{2}-4x+4}$$