Does this polynomial factorize further? I just did a national exam and this question was in it; I am convinced this does not work: > Given that $(x - 1)$ is a factor of $x^3 + 3x^2 + x - 5$, factorize this cubic fully. My attempt 1 | 1 3 1 -5 | 1 4 5 |____________ 1 4 5 0 $$(x - 1)(x^2 + 4x + 5) = 0$$ That's all I got. I gave up and couldn't factorize further. Am I missing something?

The discriminant of your polynomial is $$\Delta=4^2-4\cdot 5<0$$ so this has no real roots, that is, factorization over $\Bbb R$ is finished. On the other hand, factorization over $\Bbb C$ is possible, yielding

$$x^2+4x+5=(x+2)^2+1=(x+2-i)(x+2+i)$$