The root of polynomial $ p(x) $ is:
$$ \begin{aligned}x &= -\dfrac{ 1 }{ 6 } \end{aligned} $$Step 1:
Get rid of fractions by multipling equation by $ \color{blue}{ 2 } $.
$$ \begin{aligned} 9x+\frac{3}{2} & = 0 ~~~ / \cdot \color{blue}{ 2 } \\[1 em] 18x+3 & = 0 \end{aligned} $$Step 2:
$$ \begin{aligned} 18x+3 & = 0 \\[1 em] 18 \cdot x & = -3 \\[1 em] x & = - \frac{ 3 }{ 18 } \\[1 em] x & = - \frac{ 3 : 3 }{ 18 : 3} \\[1 em] x & = -\frac{ 1 }{ 6 } \end{aligned} $$