Ok My turn
the roots have to be factors of 6 so I'd look at 1 first
f(1)=1+4+1-6=0 therefore (x-1) is a factor [I am using factor theorem but it is just logical really]
I could now do algebraic division or synthetic division to find its cofactor but I'll just try easy possibilities first.
f(-2)=-8+16+-2-6=0 therefore (x+2) is another factor
(x-1)(x+2)=x2+x-2
$$(x^3+4x^2+x-6) \div (x^2+x-2)
=x+3$$
I did the division but I don't know how to set it out in latex. SO
$$(x^3+4x^2+x-6)=(x-1)(x+2)(x+3)$$
And that will work every time so long as the roots are integers.