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2x^{3}-\frac{3x^{2}}{2}+2x
Divide 6x^{3} by 3 to get 2x^{3}.
\frac{2\left(2x^{3}+2x\right)}{2}-\frac{3x^{2}}{2}
To add or subtract expressions, expand them to make their denominators the same. Multiply 2x^{3}+2x times \frac{2}{2}.
\frac{2\left(2x^{3}+2x\right)-3x^{2}}{2}
Since \frac{2\left(2x^{3}+2x\right)}{2} and \frac{3x^{2}}{2} have the same denominator, subtract them by subtracting their numerators.
\frac{4x^{3}+4x-3x^{2}}{2}
Do the multiplications in 2\left(2x^{3}+2x\right)-3x^{2}.
\frac{4x^{3}-3x^{2}+4x}{2}
Factor out \frac{1}{2}.
x\left(4x^{2}-3x+4\right)
Consider 4x^{3}-3x^{2}+4x. Factor out x.
\frac{x\left(4x^{2}-3x+4\right)}{2}
Rewrite the complete factored expression. Polynomial 4x^{2}-3x+4 is not factored since it does not have any rational roots.