Given that (x−1) is a factor of x^3+3x^2+x−5, factorize this cubic fully.

Given that $\left(x-1\right)$ is a factor of ${x}^{3}+3{x}^{2}+x-5$, factorize this cubic fully.
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nutnhonyl8
The discriminant of your polynomial is$\mathrm{\Delta }={4}^{2}-4\cdot 5<0$
so this has no real roots, that is, factorization over $\mathbb{R}$ is finished. On the other hand, factorization over $\mathbb{C}$ is possible, yielding
${x}^{2}+4x+5=\left(x+2{\right)}^{2}+1=\left(x+2-i\right)\left(x+2+i\right)$