It would help if you would state what you are trying to find!
Here's what I got from your work:
[imath]f_x(x, y) = 2xy + 4x \implies f_x(1, 1) = 6[/imath]
(Note: [imath]f'_x(x, y)[/imath] has no meaning. Leave the "prime" off here.)
[imath]f_y(x, y) = x^2 - 2y \implies f_y(1, 1) = -1[/imath]
We have the vector [imath]\textbf{v} = ( t^2 - t + 1, 6t + 6)[/imath]
It looks like you are asking what values of t makes [imath]\textbf{v}[/imath] perpendicular to [imath]\textbf{\nabla}f(x, y)[/imath] at x = 1, y = 1. Note that 6 - 6 = 0...
-Dan