<s>[INST] How can I find the roots of a quartic equation, knowing one of its roots? I need to decompose (in $\Bbb{C}[x]$) the function
$$
f(x) = x^4 + 4x^3 - 4x^2 + 24x + 15
$$
in its simplest form, knowing that $1 - 2i$ is one of its roots. Any ideas?
 [/INST] The answer has already been given in the comments, so I'll make a community answer out of it.
So we have the function $$f(x) = x^4 + 4x^3 - 4x^2 + 24x + 15$$
One root has been found at $x = 1-2i$.
Since the coefficients are real, there must be another root at $x = 1+2i$. We can now reduce this equation to a quadratic one, by dividing those out.
$$\begin{align}
    & \frac{x^4 + 4x^3 - 4x^2 + 24x + 15}{(x-(1-2i))(x-(1+2i))} \\
= \ & \frac{x^4 + 4x^3 - 4x^2 + 24x + 15}{(x-1+2i)(x-1-2i)} \\
= \ & \frac{x^4 + 4x^3 - 4x^2 + 24x + 15}{x^2-2x+5} \\
= \ & x^2+6x+3
\end{align}$$
The roots for that quadratic equation can be found at $x = -3-\sqrt{6}$ and $x = -3 + \sqrt{6}$.
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