How can I find the saddle point of $Z = x^2 + 3y^2 + 3xy – 6x + 3y – 6$ $$Z = x^2 + 3y^2 + 3xy  –  6x + 3y – 6$$

how can I find the maxima, minima and the saddle points of this equation?

I tried this by finding $fx, fy,fxx,fyy,fxy$
$$fx=2x+3y-6$$$$fy=6y+3x+3$$$$fxx=2$$$$fyy=6$$$$fxy=3$$

$$fx=fy=0$$
find $(x,y)$
$$x=15$$$$y=-8$$
But when I calculate $Z$ I get $Z=-63$ but in my textbook it says $Z=-51$ what am I doing wrong?
I also did the calculations, there's nothing wrong with your calculations, maybe there's a mistake in your textbook.

all your calculations are correct$$x=15$$$$y=-8$$
$$Z(15,-8)=-63$$
Your calculations are correct. I have use MatDeck to do the math and the result is the same. You can find the link for calculation in MatDeck bellow

Saddle point calculation in MatDeck