All the values of 'a' for which $\int\limits_1^2\left\{a^2+(4-4 a) x+4 x^3\right\} d x \leq 12$ are given by

a = 3

We have,

$\int\limits_1^2\left\{a^2+(4-4 a) x+4 x^3\right\} d x \leq 12$

$\Rightarrow \left[a^2+(2-2 a) x^2+x^4\right]_1^2 \leq 12$

$\Rightarrow a^2+(2-2 a)(4-1)+(16-1) \leq 12$

$\Rightarrow a^2+6-6 a+15 \leq 12$

$\Rightarrow a^2-6 a+9 \leq 0 \Rightarrow(a-3)^2 \leq 0 \Rightarrow a=3$