Prove that the expression 2x(x–6)–3(x2–4x+1) is negative for all values of x.

Simplify the following:
2 x (x - 6) - 3 (x^2 - 4 x + 1)

2 x (x - 6) = 2 x^2 - 12 x:
2 x^2 - 12 x - 3 (x^2 - 4 x + 1)

-3 (x^2 - 4 x + 1) = -3 x^2 + 12 x - 3:
-3 x^2 + 12 x - 3 + 2 x^2 - 12 x

Grouping like terms, 2 x^2 - 3 x^2 + 12 x - 12 x - 3 = (2 x^2 - 3 x^2) - 3 + (-12 x + 12 x):
(2 x^2 - 3 x^2) - 3 + (-12 x + 12 x)

2 x^2 - 3 x^2 = -x^2:
-x^2 - 3 + (-12 x + 12 x)

12 x - 12 x = 0:
-x^2 - 3

Factor -1 out of -x^2 - 3:
  -(x^2 + 3)