Solve 3x^3+5x^2-3+14x^4div(2x-1+x^2)

$$3 x ^ { 3 } + 5 x ^ { 2 } - 3 + 14 x ^ { 4 } \div ( 2 x - 1 + x ^ { 2 } )$$

Evaluate

$$\frac{3x^{5}+25x^{4}+7x^{3}-8x^{2}-6x+3}{x^{2}+2x-1}$$

Short Solution Steps

Factor 2x-1+x^{2}.

$$3x^{3}+5x^{2}-3+\frac{14x^{4}}{\left(x-\left(\sqrt{2}-1\right)\right)\left(x-\left(-\sqrt{2}-1\right)\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3x^{3}+5x^{2}-3 times \frac{\left(x-\left(\sqrt{2}-1\right)\right)\left(x-\left(-\sqrt{2}-1\right)\right)}{\left(x-\left(\sqrt{2}-1\right)\right)\left(x-\left(-\sqrt{2}-1\right)\right)}.

$$\frac{\left(3x^{3}+5x^{2}-3\right)\left(x-\left(\sqrt{2}-1\right)\right)\left(x-\left(-\sqrt{2}-1\right)\right)}{\left(x-\left(\sqrt{2}-1\right)\right)\left(x-\left(-\sqrt{2}-1\right)\right)}+\frac{14x^{4}}{\left(x-\left(\sqrt{2}-1\right)\right)\left(x-\left(-\sqrt{2}-1\right)\right)}$$

Since \frac{\left(3x^{3}+5x^{2}-3\right)\left(x-\left(\sqrt{2}-1\right)\right)\left(x-\left(-\sqrt{2}-1\right)\right)}{\left(x-\left(\sqrt{2}-1\right)\right)\left(x-\left(-\sqrt{2}-1\right)\right)} and \frac{14x^{4}}{\left(x-\left(\sqrt{2}-1\right)\right)\left(x-\left(-\sqrt{2}-1\right)\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(3x^{3}+5x^{2}-3\right)\left(x-\left(\sqrt{2}-1\right)\right)\left(x-\left(-\sqrt{2}-1\right)\right)+14x^{4}}{\left(x-\left(\sqrt{2}-1\right)\right)\left(x-\left(-\sqrt{2}-1\right)\right)}$$

Do the multiplications in \left(3x^{3}+5x^{2}-3\right)\left(x-\left(\sqrt{2}-1\right)\right)\left(x-\left(-\sqrt{2}-1\right)\right)+14x^{4}.

$$\frac{3x^{5}+6x^{4}-3x^{3}+5x^{4}+10x^{3}-5x^{2}-3x^{2}-6x+3+14x^{4}}{\left(x-\left(\sqrt{2}-1\right)\right)\left(x-\left(-\sqrt{2}-1\right)\right)}$$

Combine like terms in 3x^{5}+6x^{4}-3x^{3}+5x^{4}+10x^{3}-5x^{2}-3x^{2}-6x+3+14x^{4}.

$$\frac{3x^{5}+25x^{4}+7x^{3}-8x^{2}-6x+3}{\left(x-\left(\sqrt{2}-1\right)\right)\left(x-\left(-\sqrt{2}-1\right)\right)}$$

Expand \left(x-\left(\sqrt{2}-1\right)\right)\left(x-\left(-\sqrt{2}-1\right)\right).

$$\frac{3x^{5}+25x^{4}+7x^{3}-8x^{2}-6x+3}{x^{2}+2x-\left(\sqrt{2}\right)^{2}+1}$$

The square of \sqrt{2} is 2.

$$\frac{3x^{5}+25x^{4}+7x^{3}-8x^{2}-6x+3}{x^{2}+2x-2+1}$$

Add -2 and 1 to get -1.

$$\frac{3x^{5}+25x^{4}+7x^{3}-8x^{2}-6x+3}{x^{2}+2x-1}$$

Differentiate w.r.t. x

$$\frac{x\left(9x^{5}+74x^{4}+142x^{3}-72x^{2}-31x+10\right)}{x^{4}+4x^{3}+2x^{2}-4x+1}$$