We can factorize the quadratic as
[tex]3x^2 + 4x - 12 = 3 (x - r) (x - s)[/tex]
and expand the right side to get
[tex]3x^2 + 4x - 12 = 3x^2 - 3(r + s) x + 3rs[/tex]
[tex]\implies r + s = -\dfrac43 \text{ and } rs = -4[/tex]
Then we find
[tex](r + s)^2 = r^2 + 2rs + s^2 \implies r^2 + s^2 = \left(-\dfrac43\right)^2 - 2(-4) = \boxed{\dfrac{88}9}[/tex]
