\[\frac{3{x}^{3}-2{x}^{2}-3+4x}{3+3x+{x}^{2}}\]

1
Polynomial Division: Divide 3x32x23+4x3{x}^{3}-2{x}^{2}-3+4x by 3+3x+x23+3x+{x}^{2}.

3x3x11-11
x2+3x+3x^2+3x+33x33x^32x2-2x^24x4x3-3
3x33x^39x29x^29x9x
11x2-11x^25x-5x3-3
11x2-11x^233x-33x33-33
28x28x3030

2
Rewrite the expression using the above.
3x11+28x+30x2+3x+33x-11+\frac{28x+30}{{x}^{2}+3x+3}

Done

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