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:

Answer: - (x^2 + 3)