Solve x^2+x-6div(x^2+1)(x-1)

$${ x }^{ 2 } +x-6 \div ( { x }^{ 2 } +1)(x-1)$$

Evaluate

$$\frac{x^{4}+x^{3}+x^{2}-5x+6}{x^{2}+1}$$

Short Solution Steps

Express \frac{6}{x^{2}+1}\left(x-1\right) as a single fraction.

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

Use the distributive property to multiply 6 by x-1.

$$x^{2}+x-\frac{6x-6}{x^{2}+1}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{2}+x times \frac{x^{2}+1}{x^{2}+1}.

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

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

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

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

$$\frac{x^{4}+x^{2}+x^{3}+x-6x+6}{x^{2}+1}$$

Combine like terms in x^{4}+x^{2}+x^{3}+x-6x+6.

$$\frac{x^{4}+x^{2}+x^{3}-5x+6}{x^{2}+1}$$

Expand

$$\frac{x^{4}+x^{3}+x^{2}-5x+6}{x^{2}+1}$$

Short Solution Steps

Express \frac{6}{x^{2}+1}\left(x-1\right) as a single fraction.

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

Use the distributive property to multiply 6 by x-1.

$$x^{2}+x-\frac{6x-6}{x^{2}+1}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{2}+x times \frac{x^{2}+1}{x^{2}+1}.

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

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

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

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

$$\frac{x^{4}+x^{2}+x^{3}+x-6x+6}{x^{2}+1}$$

Combine like terms in x^{4}+x^{2}+x^{3}+x-6x+6.

$$\frac{x^{4}+x^{2}+x^{3}-5x+6}{x^{2}+1}$$