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

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

Evaluate

$$-\frac{2+4x-x^{2}}{x-2}$$

Short Solution Steps

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

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

Since \frac{x^{2}-4x-3}{x-3} and \frac{-1}{x-3} have the same denominator, subtract them by subtracting their numerators.

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

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

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

Combine like terms in x^{2}-4x-3+1.

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

Factor the expressions that are not already factored in \frac{x^{2}-2x+1}{x^{2}-3x+2}.

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

Cancel out x-1 in both numerator and denominator.

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

Since \frac{x-1}{x-2} and \frac{2}{x-2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}-4x-2}{x-3}\times \frac{x-1-2}{x-2}$$

Combine like terms in x-1-2.

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

Multiply \frac{x^{2}-4x-2}{x-3} times \frac{x-3}{x-2} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out x-3 in both numerator and denominator.

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

Expand

$$-\frac{2+4x-x^{2}}{x-2}$$

Short Solution Steps

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

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

Since \frac{x^{2}-4x-3}{x-3} and \frac{-1}{x-3} have the same denominator, subtract them by subtracting their numerators.

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

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

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

Combine like terms in x^{2}-4x-3+1.

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

Factor the expressions that are not already factored in \frac{x^{2}-2x+1}{x^{2}-3x+2}.

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

Cancel out x-1 in both numerator and denominator.

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

Since \frac{x-1}{x-2} and \frac{2}{x-2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}-4x-2}{x-3}\times \frac{x-1-2}{x-2}$$

Combine like terms in x-1-2.

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

Multiply \frac{x^{2}-4x-2}{x-3} times \frac{x-3}{x-2} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out x-3 in both numerator and denominator.

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