Solve x-7x^2/x^3+3x+4

$$x - 7 \frac { x ^ { 2 } } { x ^ { 3 } + 3 x + 4 }$$

Evaluate

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

Short Solution Steps

Express 7\times \frac{x^{2}}{x^{3}+3x+4} as a single fraction.

$$x-\frac{7x^{2}}{x^{3}+3x+4}$$

Factor x^{3}+3x+4.

$$x-\frac{7x^{2}}{\left(x+1\right)\left(x^{2}-x+4\right)}$$

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

$$\frac{x\left(x+1\right)\left(x^{2}-x+4\right)}{\left(x+1\right)\left(x^{2}-x+4\right)}-\frac{7x^{2}}{\left(x+1\right)\left(x^{2}-x+4\right)}$$

Since \frac{x\left(x+1\right)\left(x^{2}-x+4\right)}{\left(x+1\right)\left(x^{2}-x+4\right)} and \frac{7x^{2}}{\left(x+1\right)\left(x^{2}-x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\left(x+1\right)\left(x^{2}-x+4\right)-7x^{2}}{\left(x+1\right)\left(x^{2}-x+4\right)}$$

Do the multiplications in x\left(x+1\right)\left(x^{2}-x+4\right)-7x^{2}.

$$\frac{x^{4}-x^{3}+4x^{2}+x^{3}-x^{2}+4x-7x^{2}}{\left(x+1\right)\left(x^{2}-x+4\right)}$$

Combine like terms in x^{4}-x^{3}+4x^{2}+x^{3}-x^{2}+4x-7x^{2}.

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

Expand \left(x+1\right)\left(x^{2}-x+4\right).

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

Factor

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