Least value of expression x^2+4y^2+3z^2-4x-20y-12z+45 is.?

Hello,

4 is the least value.

Indeed:

x²-4x+4 = (x-2) ²

4y²-20y+25 = (2y-5) ²

3 (z²-4z+4) = 3 (z-2) ²

x²+4y²+3z²-4x-20y-12z+45 = (x-2) ² + (2y-5) ²+3 (z-2) ²+45-4-25-12

= (x-2) ² + (2y-5) ²+3 (z-2) ²+4

Minimum is 4 when x=2, y=5/2 and z=2 for square is always positive.