[tex]\bf 6x^3-11x^2-9x-1=0
\qquad
\begin{array}{lllrrrll}
-\frac{1}{2}&|&6&-11&-9&-1\\
&|&&-3&7&1\\
\textendash\textendash\textendash&|&\textendash\textendash\textendash&\textendash\textendash\textendash&\textendash\textendash\textendash&\textendash\textendash\textendash\\
&&6&-14&-2&0
\end{array}
\\\\
\\\\
x=-\frac{1}{2}\implies x+\frac{1}{2}=0\implies
\left( x+\frac{1}{2} \right)(6x^2-14x-2)=0
\\\\
\textit{taking common factor on the quadratic}
\\\\
[/tex][tex]\bf \left( x+\frac{1}{2} \right)2(3x^2-7x-1)=0\implies \left( x+\frac{1}{2} \right)(3x^2-7x-1)=0 \\\\ \textit{doing the quadratic formula on it} \\\\ x=\cfrac{-(-7)\pm\sqrt{(-7)^2-4(3)(-1)}}{2(3)} \\\\\\ x=\cfrac{7\pm\sqrt{49+12}}{6}\implies x=\cfrac{7\pm\sqrt{61}}{6}\impliedby \textit{61 is a prime number} \\\\\\ thus\implies x\to \begin{cases} -\frac{1}{2} \\\\ \cfrac{7+\sqrt{61}}{6} \\\\ \cfrac{7-\sqrt{61}}{6} \end{cases}[/tex]
