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

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

Evaluate

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

Short Solution Steps

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

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

Since \frac{\left(x-1\right)\left(x^{2}+1\right)}{x^{2}+1} and \frac{5}{x^{2}+1} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(x-1\right)\left(x^{2}+1\right)+5.

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

Combine like terms in x^{3}+x-x^{2}-1+5.

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

Differentiate w.r.t. x

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