The point $\left(a^2,a\right)$ lies between the straight lines $x+y=6\text{ and }x+y=2$ for 

$\left(a^2,a\right)$ lies between the given lines

if $a^2+a-6>0\text{ and }{a}^2+a-2<0$            ..(1)

or $a^2+a-6<0\text{ and }{a}^2+a-2>0$           ..(2)

(1) does not hold for any value of $a$

From $\text{ (2) }-3<a<2\text{ and }[a<-2\text{ or }a>1]$

$\begin{array}{l}\Rightarrow -3<a<-2\text{ or }1\mid <a<2\\ \Rightarrow |2a+5|<1\text{ or }|2a-3|<1\end{array}$