<s>[INST] How do you divide a polynomial by a binomial of the form $ax^2+b$, where $a$ and $b$ are greater than one? I came across a question that asked me to divide $-2x^3+4x^2-3x+5$ by $4x^2+5$. Can anyone help me?
 [/INST] I will try this way:
Since you are dividing a 3rd degree polynomial by a 2nd degree polynomial, WLOG, we may assume
$$-2x^3+4x^2-3x+5=(4x^2+5)(ax+b)+cx+d\quad(1)$$
Now, comparing the coefficients of $x^3$ and $x^2$ readily give $a=-\frac{1}{2}$ and $b=1$. Comparing coefficients of $x$, we have $5a+c=-3\Rightarrow c=-\frac{1}{2}$. Finally comparing the constant term, we get $5b+d=5\Rightarrow d=0$.
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