<s>[INST] Algebra Problem The expression $x^2-4x+5$ is a factor of $ax^3+bx^2+25$. Express the sum a+b as an integer. 
Please give an explanation of how the answer 
 [/INST] Since $x^2-4x+5=0$ has the roots $2 \pm i$ they must also be roots of $ax^3+bx^2+25\,$ therefore:
$$
0 = a(2+i)^3+b(2+i)^2+25=8a -6a + 4b - b + 25 + i\,(12 a -a + 4b) \\
\iff
\begin{cases}
\begin{align}
2a + 3b + 25 = 0 \\
11 a + 4b = 0
\end{align}
\end{cases}
$$
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