HMMT November 2017 T8

Find the number of quadruples $(a, b, c, d)$ of integers with absolute value at most 5 such that

$$\left(a^{2}+b^{2}+c^{2}+d^{2}\right)^{2}=(a+b+c+d)(a-b+c-d)\left((a-c)^{2}+(b-d)^{2}\right)$$