Solution:

a) The discriminant of $f(x)$ is equal to $4a^{2} + 13 > 0$.

b) We consecutively have
$$\begin{aligned} -72 & = x_{1}^{3} + x_{2}^{3} = (x_{1} + x_{2}) \left[ (x_{1} + x_{2})^{2} - 3 x_{1} x_{2} \right] \\ & = (1 - 2a) \left( 4a^{2} - a + 10 \right ) = -8a^{3} + 6a^{2} - 21a + 10 \end{aligned}$$
Therefore the required values of $a$ are the real solutions of the equation
$$8a^{3} - 6a^{2} + 21a - 82 = 0$$
Since $a = 2$ is a solution of this equation, we obtain
$$8a^{3} - 6a^{2} + 21a - 82 = (a - 2)(8a^{2} + 10a + 41)$$
The equation $8a^{2} + 10a + 41 = 0$ has no real roots. So the only solution of the problem is $a = 2$.