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

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

Evaluate

$$\frac{8x^{3}-24x^{2}-13x+4}{\left(x-4\right)\left(x-3\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-4}{x-4}.

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

Since \frac{x-4}{x-4} and \frac{x-2}{x-4} have the same denominator, add them by adding their numerators.

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

Combine like terms in x-4+x-2.

$$\frac{4x^{2}-1}{x-3}+\left(x+1\right)\times \frac{x}{x-2}\left(1+\frac{x-2}{x-4}+\frac{x-1}{x-3}\times \frac{2x-6}{x-4}\right)$$

Multiply \frac{x-1}{x-3} times \frac{2x-6}{x-4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{4x^{2}-1}{x-3}+\left(x+1\right)\times \frac{x}{x-2}\left(1+\frac{x-2}{x-4}+\frac{\left(x-1\right)\left(2x-6\right)}{\left(x-3\right)\left(x-4\right)}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-4}{x-4}.

$$\frac{4x^{2}-1}{x-3}+\left(x+1\right)\times \frac{x}{x-2}\left(\frac{x-4}{x-4}+\frac{x-2}{x-4}+\frac{\left(x-1\right)\left(2x-6\right)}{\left(x-3\right)\left(x-4\right)}\right)$$

Since \frac{x-4}{x-4} and \frac{x-2}{x-4} have the same denominator, add them by adding their numerators.

$$\frac{4x^{2}-1}{x-3}+\left(x+1\right)\times \frac{x}{x-2}\left(\frac{x-4+x-2}{x-4}+\frac{\left(x-1\right)\left(2x-6\right)}{\left(x-3\right)\left(x-4\right)}\right)$$

Combine like terms in x-4+x-2.

$$\frac{4x^{2}-1}{x-3}+\left(x+1\right)\times \frac{x}{x-2}\left(\frac{2x-6}{x-4}+\frac{\left(x-1\right)\left(2x-6\right)}{\left(x-3\right)\left(x-4\right)}\right)$$

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

$$\frac{4x^{2}-1}{x-3}+\left(x+1\right)\times \frac{x}{x-2}\left(\frac{\left(2x-6\right)\left(x-3\right)}{\left(x-4\right)\left(x-3\right)}+\frac{\left(x-1\right)\left(2x-6\right)}{\left(x-4\right)\left(x-3\right)}\right)$$

Since \frac{\left(2x-6\right)\left(x-3\right)}{\left(x-4\right)\left(x-3\right)} and \frac{\left(x-1\right)\left(2x-6\right)}{\left(x-4\right)\left(x-3\right)} have the same denominator, add them by adding their numerators.

$$\frac{4x^{2}-1}{x-3}+\left(x+1\right)\times \frac{x}{x-2}\times \frac{\left(2x-6\right)\left(x-3\right)+\left(x-1\right)\left(2x-6\right)}{\left(x-4\right)\left(x-3\right)}$$

Do the multiplications in \left(2x-6\right)\left(x-3\right)+\left(x-1\right)\left(2x-6\right).

$$\frac{4x^{2}-1}{x-3}+\left(x+1\right)\times \frac{x}{x-2}\times \frac{2x^{2}-6x-6x+18+2x^{2}-6x-2x+6}{\left(x-4\right)\left(x-3\right)}$$

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

$$\frac{4x^{2}-1}{x-3}+\left(x+1\right)\times \frac{x}{x-2}\times \frac{4x^{2}-20x+24}{\left(x-4\right)\left(x-3\right)}$$

Factor the expressions that are not already factored in \frac{4x^{2}-20x+24}{\left(x-4\right)\left(x-3\right)}.

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

Cancel out x-3 in both numerator and denominator.

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

Express \left(x+1\right)\times \frac{x}{x-2} as a single fraction.

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

Multiply \frac{\left(x+1\right)x}{x-2} times \frac{4\left(x-2\right)}{x-4} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out x-2 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-3 and x-4 is \left(x-4\right)\left(x-3\right). Multiply \frac{4x^{2}-1}{x-3} times \frac{x-4}{x-4}. Multiply \frac{4x\left(x+1\right)}{x-4} times \frac{x-3}{x-3}.

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

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

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

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

$$\frac{4x^{3}-16x^{2}-x+4+4x^{3}-12x^{2}+4x^{2}-12x}{\left(x-4\right)\left(x-3\right)}$$

Combine like terms in 4x^{3}-16x^{2}-x+4+4x^{3}-12x^{2}+4x^{2}-12x.

$$\frac{8x^{3}-24x^{2}-13x+4}{\left(x-4\right)\left(x-3\right)}$$

Expand \left(x-4\right)\left(x-3\right).

$$\frac{8x^{3}-24x^{2}-13x+4}{x^{2}-7x+12}$$

Expand

$$\frac{8x^{3}-24x^{2}-13x+4}{\left(x-4\right)\left(x-3\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-4}{x-4}.

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

Since \frac{x-4}{x-4} and \frac{x-2}{x-4} have the same denominator, add them by adding their numerators.

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

Combine like terms in x-4+x-2.

$$\frac{4x^{2}-1}{x-3}+\left(x+1\right)\times \frac{x}{x-2}\left(1+\frac{x-2}{x-4}+\frac{x-1}{x-3}\times \frac{2x-6}{x-4}\right)$$

Multiply \frac{x-1}{x-3} times \frac{2x-6}{x-4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{4x^{2}-1}{x-3}+\left(x+1\right)\times \frac{x}{x-2}\left(1+\frac{x-2}{x-4}+\frac{\left(x-1\right)\left(2x-6\right)}{\left(x-3\right)\left(x-4\right)}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-4}{x-4}.

$$\frac{4x^{2}-1}{x-3}+\left(x+1\right)\times \frac{x}{x-2}\left(\frac{x-4}{x-4}+\frac{x-2}{x-4}+\frac{\left(x-1\right)\left(2x-6\right)}{\left(x-3\right)\left(x-4\right)}\right)$$

Since \frac{x-4}{x-4} and \frac{x-2}{x-4} have the same denominator, add them by adding their numerators.

$$\frac{4x^{2}-1}{x-3}+\left(x+1\right)\times \frac{x}{x-2}\left(\frac{x-4+x-2}{x-4}+\frac{\left(x-1\right)\left(2x-6\right)}{\left(x-3\right)\left(x-4\right)}\right)$$

Combine like terms in x-4+x-2.

$$\frac{4x^{2}-1}{x-3}+\left(x+1\right)\times \frac{x}{x-2}\left(\frac{2x-6}{x-4}+\frac{\left(x-1\right)\left(2x-6\right)}{\left(x-3\right)\left(x-4\right)}\right)$$

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

$$\frac{4x^{2}-1}{x-3}+\left(x+1\right)\times \frac{x}{x-2}\left(\frac{\left(2x-6\right)\left(x-3\right)}{\left(x-4\right)\left(x-3\right)}+\frac{\left(x-1\right)\left(2x-6\right)}{\left(x-4\right)\left(x-3\right)}\right)$$

Since \frac{\left(2x-6\right)\left(x-3\right)}{\left(x-4\right)\left(x-3\right)} and \frac{\left(x-1\right)\left(2x-6\right)}{\left(x-4\right)\left(x-3\right)} have the same denominator, add them by adding their numerators.

$$\frac{4x^{2}-1}{x-3}+\left(x+1\right)\times \frac{x}{x-2}\times \frac{\left(2x-6\right)\left(x-3\right)+\left(x-1\right)\left(2x-6\right)}{\left(x-4\right)\left(x-3\right)}$$

Do the multiplications in \left(2x-6\right)\left(x-3\right)+\left(x-1\right)\left(2x-6\right).

$$\frac{4x^{2}-1}{x-3}+\left(x+1\right)\times \frac{x}{x-2}\times \frac{2x^{2}-6x-6x+18+2x^{2}-6x-2x+6}{\left(x-4\right)\left(x-3\right)}$$

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

$$\frac{4x^{2}-1}{x-3}+\left(x+1\right)\times \frac{x}{x-2}\times \frac{4x^{2}-20x+24}{\left(x-4\right)\left(x-3\right)}$$

Factor the expressions that are not already factored in \frac{4x^{2}-20x+24}{\left(x-4\right)\left(x-3\right)}.

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

Cancel out x-3 in both numerator and denominator.

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

Express \left(x+1\right)\times \frac{x}{x-2} as a single fraction.

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

Multiply \frac{\left(x+1\right)x}{x-2} times \frac{4\left(x-2\right)}{x-4} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out x-2 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-3 and x-4 is \left(x-4\right)\left(x-3\right). Multiply \frac{4x^{2}-1}{x-3} times \frac{x-4}{x-4}. Multiply \frac{4x\left(x+1\right)}{x-4} times \frac{x-3}{x-3}.

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

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

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

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

$$\frac{4x^{3}-16x^{2}-x+4+4x^{3}-12x^{2}+4x^{2}-12x}{\left(x-4\right)\left(x-3\right)}$$

Combine like terms in 4x^{3}-16x^{2}-x+4+4x^{3}-12x^{2}+4x^{2}-12x.

$$\frac{8x^{3}-24x^{2}-13x+4}{\left(x-4\right)\left(x-3\right)}$$

Expand \left(x-4\right)\left(x-3\right).

$$\frac{8x^{3}-24x^{2}-13x+4}{x^{2}-7x+12}$$