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\left(x+3\right)\left(-x^{2}+3x-2\right)
By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -6 and q divides the leading coefficient -1. One such root is -3. Factor the polynomial by dividing it by x+3.
a+b=3 ab=-\left(-2\right)=2
Consider -x^{2}+3x-2. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-2. To find a and b, set up a system to be solved.
a=2 b=1
Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.
\left(-x^{2}+2x\right)+\left(x-2\right)
Rewrite -x^{2}+3x-2 as \left(-x^{2}+2x\right)+\left(x-2\right).
-x\left(x-2\right)+x-2
Factor out -x in -x^{2}+2x.
\left(x-2\right)\left(-x+1\right)
Factor out common term x-2 by using distributive property.
\left(x-2\right)\left(-x+1\right)\left(x+3\right)
Rewrite the complete factored expression.