<s>[INST] Is it possible to solve the following equation without using the Rational Root Theorem? Given $f(x)=x^4+2x^3+2x^2-2x-3$, where $x-1$ is a factor of $f(x)$, how is it possible to solve $f(x)$ without the Rational Root Theorem?
Here's my progress:
$$f(x)=x^4+2x^3+2x^2-2x-3$$
$$f(x)=(x-1)(x^3+3x^2-5x+3)$$
And there's where I got stuck.
I cheated, though, and tried solving this equation with the Rational Root Theorem. That's what I got:
$$f(x)=x^4+2x^3+2x^2-2x-3$$
$$f(x)=(x-1)(x^3+3x^2-5x+3)$$
$$f(x)=(x-1)(x+1)(x^2+2x+3)$$
$$f(x)=(x-1)(x+1)(x-(-1-\sqrt2))(x-(-1+\sqrt2))$$
Please, if you're answering this question, I'd really appreciate if you could explain your procedures and what technique you used.
Thank you very much.
 [/INST] $$\begin{aligned}
x^4+2x^3+2x^2-2x-3
&=(x^4+2x^2-3)+(2x^3-2x) \\
&=(x^2+3)(x^2-1)+2x(x^2-1) \\
&=(x^2-1)(x^2+3+2x) \\
&=(x^2-1)(x^2+2x+3)
\end{aligned}$$
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