Minimum Value of Quadratic Expression

The term $4(y - 3)^2 \ge 0$ independently, so its minimum value is $0$ at $y = 3$. Now consider the $x$-dependent part: $E(x) = 4(x - 2)^2 - 2(x - 3)^2 = 4(x^2 - 4x + 4) - 2(x^2 - 6x + 9)$ $= 4x^2 - 16x + 16 - 2x^2 + 12x - 18 = 2x^2 - 4x - 2$ $= 2(x^2 - 2x + 1) - 4 = 2(x - 1)^2 - 4$.

The minimum value of $2(x - 1)^2 - 4$ occurs at $x = 1$, giving $-4$. Therefore, the least possible total value is $-4 + 0 = -4$.