Find the probability for having real roots with the polynomial $x^2+ax+b$

Assuming that the distribution is uniform, you just need to compute the ratio between the shadowed area in the picture and the area of the rectangle.

Looking at $a,b$ as random variables, you just have that \begin{align*} P(a^2-4c>0)=& \frac{1}{6} \left(3+ \int_1^3 \frac 14 a^2 da \right)=\frac{43}{48} \end{align*}