Answer
$\{a_0:\text{$a_0$ is real}\}$.
Work Step by Step
The kernel of a linear map $T : A → B$, is the set of all elements $v$ in $A$ for which $T(v) = 0$, where $0$ is the zero vector in B.
Hence, here we need $a_1+2a_2x=0$, so $a_1=a_2=0$, and the others are free variables. Thus the kernel is $\{a_0:\text{$a_0$ is real}\}$.
