Divide the polynomials -13x^2+4x^3+2x-7 and explain each step. State the answer in full sentence and ecxpress it as one expression.

Divide the polynomials $$\displaystyle-{13}{x}^{{2}}+{4}{x}^{{3}}+{2}{x}-{7}$$ and explain each step. State the answer in full sentence and ecxpress it as one expression. Check your solution.

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Given:
To divide the polynomials $$\displaystyle{\frac{{{4}{x}^{{3}}-{13}{x}^{{2}}+{2}{x}-{7}}}{{{x}^{{2}}+{3}{x}-{2}}}}$$ and verify the solution.
To divide the polynomials, first find the quotient by dividing the higher degree of each polynomial.
$$\displaystyle{\frac{{{4}{x}^{{3}}}}{{{x}^{{2}}}}}={4}{x}$$
Let's divide the polynomial with quotient 4x.
$$\displaystyle{Q}={4}{x},{R}=-{25}{x}^{{2}}+{10}{x}-{7}$$
Thus it can be expressed as,
Now, apply it with the expressions obtained above.
$$\displaystyle{4}{x}^{{3}}-{13}{x}^{{2}}+{2}{x}-{7}={4}{x}-{15}{x}^{{2}}+{3}{x}-{2}+{\left(-{5}{x}\right)}-{37}$$
Let's prove the result using the single term expressed.
$$\displaystyle{4}{x}^{{3}}-{13}{x}^{{2}}+{2}{x}-{7}={\left({4}{x}-{25}\right)}{\left({x}^{{2}}+{3}{x}-{2}\right)}+{\left({85}{x}-{57}\right)}$$
$$\displaystyle{4}{x}^{{3}}-{13}{x}^{{2}}+{2}{x}-{7}={4}{x}^{{3}}+{12}{x}^{{2}}-{8}{x}-{25}{x}^{{2}}-{75}{x}+{50}+{85}{x}-{57}$$
$$\displaystyle{4}{x}^{{3}}-{13}{x}^{{2}}+{2}{x}-{7}={4}{x}^{{3}}-{13}{x}^{{2}}+{2}{x}-{7}$$
LHS=RHS
Hence answer is verified.