Mid-point P of line segment  joining $(-3,-3,4)$ and $(3, 7, 6)$ is $\left(\frac{0}{2},\frac{4}{2},\frac{10}{2}\right)$ i.e., $(0,2,5)$ Equation of normal to the plane is

$\begin{array}{l}\overrightarrow{n}=(0-(-3\left)\right)\widehat{i}+(2-(-3\left)\right)\widehat{j}+(5-4)\widehat{k}\\ \Rightarrow \overrightarrow{n}=3\widehat{i}+5\widehat{j}+\widehat{k}\end{array}$

Now, equation of plane passing through (0, 2, 5) is 3(x - 0) + 5(y - 2)+1 (z - 5) =0

$\Rightarrow 3x+5y+z=15$                                                       …(i)

Clearly, (i) passes through point (4, 1,-2).