The equation of the plane through (4,4,0) and perpendicular to the planes 2x+y+2z+3=0,and  3x+3y+2z-8=0

Equation of a plane through (4,4,0) is a(x-4)+b(y-4)+c(z-0)=0 $\to \left(1\right)$

If plane (1) is perpendicular to 2x+y+2z+3=0 then 2a+b+2c=0$\to \left(2\right)$

If plane (1) is perpendicular to 3x+3y+2z-8=0 then 3a+3b+2c=0 $\to \left(3\right)$

From (2) and (3)

     a       b       c

  1     2     2      1

  3     2     3      3

$\Rightarrow \frac{a}{2-6}=\frac{b}{6-4}=\frac{c}{6-3}$

$\Rightarrow \frac{a}{4}=\frac{b}{-2}=\frac{c}{-3}$

The equation of the plane is 4x-2y-3z-8=0