<s>[INST] Find the number of possible values of $a$ 
Positive integers $a, b, c$, and $d$ satisfy $a > b > c > d, a + b + c + d = 2010$,
  and $a^2 − b^2 + c^2 − d^2 = 2010$. Find the number of possible values of $a.$

Obviously, factoring,
$$(a-b)(a+b) + (c-d)(c+d) = 2010$$ 
$$a + b + c + d = 2010$$
Substituting you get:
$$(a-b)(a+b) + (c-d)(c+d) - (a + b) - (c + d) = 0$$
$$\implies (a - b - 1)(a + b) + (c - d- 1)(c+d) = 0$$ 
But I dont see anything else.
 [/INST] Hint: examine the conditions you are given in the question - what can you say about the value of the expression in your last equation? Can you find a condition which makes it zero?
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