You know the steps and notation so I am going omit much of the notation for simplicity:
1) $8=8x\lambda$
2) $8=8y\lambda$
3) $5=10z\lambda$
multiplying equations 1 and 2 by 5 and equation 3 by 8 gives and diving out by 8 gives:
$x=y=2z$
Now replace $x$ and $y$ with $2z$ plug them into the constraint:
$z=\pm1$
now since put $\frac{1}{2}x$ or $y$ (since the are the same) in for z and solve for x:
$4x^2+4y^2+\frac{5}{2}x^2=37$
$x=y= \pm \frac{\sqrt{74}}{\sqrt{21}}$
I just don't think something is right about my x and y's given that these are functions from a textbook, answers like these are usually a red flag, I have a final in two days. I know thata after this step I take all my triplets and plug them into f(x) to find the maximum and minimum values from the function
$$x=y=2z$$
If $z=\pm 1$
Then we should have $x=y=2z=\pm 2$
A mistake is
$$4x^2+4y^2+\frac52 x^2 = 37$$
where it should have been $$4x^2+4y^2+\frac54 x^2 = 37$$
where you have forgotten to square the $\frac12$.