<s>[INST] Factoring $(x+a)(x+b)(x+c)+(b+c)(c+a)(a+b)$ and use the result to solve an equation I managed to prove that $(x+a+b+c)$ is a factor of
$$(x+a)(x+b)(x+c)+(b+c)(c+a)(a+b)$$
Then I was asked to use the result to solve
$$(x+2)(x-3)(x-1)+4=0$$
I know by comparison, $a=2, b=-3, c=-1$, and thus $(x-2)$ is a factor, but I can't really figure out how to solve the equation without expanding the brackets.
 [/INST] Unfortunately, you do have to expand the brackets.
Fortunately, it's not so messy after all:
$$(x+2)(x-3)(x-1)+4=0$$
so 
$$x^3-2x^2-5x+10=0$$
and factoring our $x-2$, 
$$(x-2)(x^2-5)=0$$
from which we can readily read the answer as $x=2,\pm\sqrt{5}$
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