A force is given by $F = at + b{t^2}$, where $t$ is time, the dimensions of $a$ and $b$ are respectively :

$\Rightarrow \$ Since $F=at+bt^2---(1)$

$\therefore$ Dimension of $at$ and $bt^2$ must be equal to force only.

Hence $[F]=[M^{1} L^{1} T^{-2}]---(2)$

$F$ from $(1)$ and $(2)$

$\Rightarrow \ [at]=a[T] =[F]$

$\therefore \ a[T] =[M^{1} L^{1} T^{-2}]$

$\therefore \ [a]=[M^{1} L^{1} T^{-3}]$

and per $bt^2=b[T^{2}]=[M^{1} L^{1} T^{-2}]$

$\therefore \ \boxed {[b]= [M^{1} L^{1} T^{-4}]}$