We know that the angle between the planes $a_{1} x+b_{1} y$ $+c_{1} z+d_{1}=0$ and $a_{2} x+b_{2} y+c_{2} z+d_{2}=0$ is given by
$\cos \theta=\left|\frac{a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2} \sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}}\right|$
Given equation of planes are $p x+2 y+2 z-3=0$ and $2 x-y+z+2=0$
On comparing with standard equations, we get
$a_{1}=p, a_{2}=2, b_{1}=2, b_{2}=-1, \quad c_{1}=2, c_{2}=1$
Also, $\theta=\frac{\pi}{4} \quad$ (given)
$\cos \frac{\pi}{4}=\left|\frac{p \times 2+2 \times(-1)+2 \times 1}{\sqrt{p^{2}+4+4} \sqrt{4+1+1}}\right|$
$\Rightarrow \frac{1}{\sqrt{2}}=\frac{2 p}{\sqrt{p^{2}+8} \sqrt{6}} \Rightarrow \frac{1}{2}=\frac{4 p^{2}}{\left(p^{2}+8\right) 6}$
$\Rightarrow \frac{3}{4}=\frac{p^{2}}{p^{2}+8}$
$\Rightarrow 3 p^{2}+24=4 p^{2} \Rightarrow p^{2}=24$