Solve 4x^2-5/x-2-x/x-2-1+x/2-x

$$4 \frac{ { x }^{ 2 } -5 }{ x-2 } - \frac{ x }{ x-2 } - \frac{ 1+x }{ 2-x }$$

Evaluate

$$\frac{4x^{2}-19}{x-2}$$

Short Solution Steps

Express 4\times \frac{x^{2}-5}{x-2} as a single fraction.

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

Since \frac{4\left(x^{2}-5\right)}{x-2} and \frac{x}{x-2} have the same denominator, subtract them by subtracting their numerators.

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

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

$$\frac{4x^{2}-20-x}{x-2}-\frac{1+x}{2-x}$$

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

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

Since \frac{4x^{2}-20-x}{x-2} and \frac{-\left(1+x\right)}{x-2} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 4x^{2}-20-x-\left(-\left(1+x\right)\right).

$$\frac{4x^{2}-20-x+1+x}{x-2}$$

Combine like terms in 4x^{2}-20-x+1+x.

$$\frac{4x^{2}-19}{x-2}$$

Expand

$$-\frac{19-4x^{2}}{x-2}$$

Short Solution Steps

Express 4\times \frac{x^{2}-5}{x-2} as a single fraction.

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

Since \frac{4\left(x^{2}-5\right)}{x-2} and \frac{x}{x-2} have the same denominator, subtract them by subtracting their numerators.

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

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

$$\frac{4x^{2}-20-x}{x-2}-\frac{1+x}{2-x}$$

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

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

Since \frac{4x^{2}-20-x}{x-2} and \frac{-\left(1+x\right)}{x-2} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 4x^{2}-20-x-\left(-\left(1+x\right)\right).

$$\frac{4x^{2}-20-x+1+x}{x-2}$$

Combine like terms in 4x^{2}-20-x+1+x.

$$\frac{4x^{2}-19}{x-2}$$