Given: $f\left(x\right)=\left|2x^2+5\left|x\right|-3\right|$

Now, let $y=\left|2x^2+5x-3\right|$

$\Rightarrow y=\left|2x^2+6x-x-3\right|$

$\Rightarrow y=\left|2x\left(x+3\right)-\left(x+3\right)\right|$

$\Rightarrow y=\left|\left(2x-1\right)\left(x+3\right)\right|$

Graph of $\Rightarrow y=\left|\left(2x-1\right)\left(x+3\right)\right|$ is given below:

Graph of $f\left(x\right)=\left|2x^2+5\left|x\right|-3\right|$ is given below:

So, $f\left(x\right)$ is continuous for $x\in R$ and non-differentiable at $x=\pm \frac{1}{2},0$

So, number of points of discontinuity$=0=m$

Also, number of points of non-discontinuity$=3=n$

$\Rightarrow m+n=3$