Simplify third degree polynomial equations. Given an equation: 

$6x^3 - 13x^2 + 8x + 3 = 0$

Broken down to get one form

$(x + {3\over2})$

How can you divide the prior equation to know it will simplify to

$(x + {3\over2})(6x^2 + 4x + 2) = 0$


It looks like you have typo. If you take the polynomial $$p(x)=6 x^3 + 13 x^2 + 8 x + 3,$$ then $-3/2$ is a root. 
This guarantees that $x+3/2$ is a factor of $p$. This follows easily from the Division Algorithm: if you write $$p=(x+3/2) q(x)+r(x),$$ with $\deg r<\deg (x+3/2)=1$, then $r$ is constant; from $p(3/2)=0$ it follows that $r=0$. So now you want to find $q$. You can do it by long division, or simply writing $q(x)=ax^2+bx+c$, and then you have 
\begin{align}
6 x^3 + 13 x^2 + 8 x + 3&=(x+3/2)(ax^2+bx+c)\\ 
&=ax^3+(3a/2+b)x^2+(c+3b/2)x+3c/2.
\end{align}
So $3c/2=3$, and $c=2$. Then $2+3b/2=8$, and $b=4$; finally, $3a/2+4=13$, and $a=6$. Thus
$$
q(x)=6x^2+4x+2.
$$
