The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= -\frac{ 5 }{ 4 }+\frac{\sqrt{ 47 }}{ 4 }i\\[1 em]x_2 &= -\frac{ 5 }{ 4 }- \frac{\sqrt{ 47 }}{ 4 }i \end{aligned} $$The solutions of $ 2x^2+5x+9 = 0 $ are: $ x = -\dfrac{ 5 }{ 4 }+\dfrac{\sqrt{ 47 }}{ 4 }i ~ \text{and} ~ x = -\dfrac{ 5 }{ 4 }-\dfrac{\sqrt{ 47 }}{ 4 }i$.
You can use step-by-step quadratic equation solver to see a detailed explanation on how to solve this quadratic equation.