Prove that (4x–5)^2 – 5x(3x – 8) is positive for all values of x.

To begin we need to simplify the expression. First we multiply out (4x–5)^2 to get 16x²+40x+25 and then we multiply out 5x(3x – 8) to get 15x²-40x. This makes the whole expression 16x²+40x+25-(15x²-40x), which equals 16x²+40x+25-15x²+40x. This simplifies to x²+25. We know that x² is positive for all values of x, and so x²+25 must also be positive for all values of x.