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.

Denote $$b=a-x, \quad d=c-y. \qquad(x,y\in\mathbb{N}).\tag{1}$$
Then
$$
a+b+c+d = 2a-x+2c-y = 2010,\\
a^2-b^2+c^2-d^2=(2a-x)x+(2c-y)y=2010.\tag{2}
$$
Since $x,y\in\mathbb{N}$, then unique solution of $(2)$ has $$x=y=1.\tag{3}$$
So, solution has form
$$
(a,\;a-1,\;c,\;c-1).\tag{4}
$$
$(2),(4)\Rightarrow$
$$a+c-1=1005,$$
$$c=1006-a.$$
So, solution has form
$$
(a,\;a-1,\;1006-a,\;1005-a).\tag{5}
$$
Smallest possible $a$: $a_{min}=504 \rightarrow (a,b,c,d)=(504,503,502,501)$.
Largest possible $a$: $a_{max}=1004 \rightarrow (a,b,c,d)=(1004,1003,2,1)$.
Number of possible values of $a$: $501$.
