Key idea. Turn each variable’s terms into a constant minus a square.

1. Complete the squares separately:

   \[\begin{aligned}20x-2x^2&=50-2(x-5)^2,\\-4y-4y^2&=1-4\left(y+\frac12\right)^2,\\6z-3z^2&=3-3(z-1)^2.\end{aligned}\]

2. After including the final \(-2\), the whole expression is

   \[52-2(x-5)^2-4\left(y+\frac12\right)^2-3(z-1)^2.\]

3. Each subtracted square is non-negative, so the expression cannot exceed 52.

4. Set \(x=5\), \(y=-1/2\), \(z=1\). All three squares vanish, and the value is 52. Hence 52 is the greatest possible value.

Remember. A maximum can be proved by writing the expression as a constant minus non-negative terms.