Is Spivak wrong here, or am I just missing something? Chapter 1 Problem 18 has the reader doing various proofs with second-degree polynomial functions of the form $x^2 + bx + c$. My issue lies with problem 18d, but it uses knowledge from 18b and 18c, so bear with me.
Part B has the reader prove that 
if $b^2 - 4c < 0$ 
then 
$x^2 + bx + c > 0$ for all x
which was easy enough to show.
Part C has the reader prove that 
if x and y are both not 0, then $x^2 + xy + y^2 > 0$
which was accomplished by substituting $y = b$ and $y^2 = c$ to get 
$x^2 + bx + c$ 
and then following the same logic of showing that because $b^2 - 4c = y^2 - 4y^2 < 0$ then $x^2 + xy + y^2 > 0$.
Then we get to 18d, where the issue is. The question is:

18 (d) For which numbers $\alpha$ is it true that $x^2 + \alpha xy + y^2 > 0$, assuming x and y are both not 0?

from the answer book it says that $\alpha$ must satisfy $(\alpha y)^2 - 4y^2 < 0$ (taken from part B, by substituting  $b$ for $\alpha y$ and c for $y^2$)
$\Rightarrow$ $\alpha ^2 < 4$ $\Rightarrow |\alpha| < 2$
but this is clearly not true for many, many examples.
Say for instance that we let x = 2 and y = 3 in our equation $x^2 + \alpha xy + y^2$ then any positive $\alpha$ would make $x^2 + \alpha xy + y^2 > 0$ be true. $\alpha$ could equal 612 and the inequality would hold true in that situation.
I think the error in logic comes from Spivak's statement that 

$\alpha$ must satisfy $(\alpha y)^2 - 4y^2 < 0$

because it assumes that it's the only condition in which the polynomial could be positive, but it's clearly not the case. Also, it's easy to solve for the inequality because we can just do some basic manipulation to come up with the solution for $\alpha$
if

$x^2 + \alpha xy + y^2 > 0$

then 

$ \alpha xy > -(x^2 + y^2 )$

then

$\alpha > -(\frac{x^2 + y^2}{xy})$

In any case, let me know if there's any faults in my logic, or is there something I'm missing.

Your logic is wrong. The problem asked for which $\alpha$'s such that: $x^2+\alpha xy + y^2 > 0$. You interpret it as: which $x,y$ such that there is an $\alpha$ such that : $x^2+\alpha xy + y^2 > 0$. These are non-equivalent statements. 
