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\left(2x+1\right)\left(x^{2}-4x+3\right)
By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 3 and q divides the leading coefficient 2. One such root is -\frac{1}{2}. Factor the polynomial by dividing it by 2x+1.
a+b=-4 ab=1\times 3=3
Consider x^{2}-4x+3. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+3. To find a and b, set up a system to be solved.
a=-3 b=-1
Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.
\left(x^{2}-3x\right)+\left(-x+3\right)
Rewrite x^{2}-4x+3 as \left(x^{2}-3x\right)+\left(-x+3\right).
x\left(x-3\right)-\left(x-3\right)
Factor out x in the first and -1 in the second group.
\left(x-3\right)\left(x-1\right)
Factor out common term x-3 by using distributive property.
\left(x-3\right)\left(x-1\right)\left(2x+1\right)
Rewrite the complete factored expression.