Prove that `-2x^2+8x-9lt0` for all real `x.`

Hello!

Let's transform the original expression (make a perfect square):

`-2x^2+8x-9 = -2(x^2-4x+9/2) = -2((x-2)^2-4+9/2) =`

`= -2((x-2)^2+1/2).`

The expression `(x-2)^2` is always nonnegative as a square of any real number. Therefore `(x-2)^2+1/2` is always positive and  `-2((x-2)^2+1/2)` is always negative. So we proved the desired result.