Answer:
This one is tricky! Let x be any integer. We can make any other integer we want by taking Ax + B, where A and B are integers too. The square of this integer is than (Ax+B)2. When expanded, this comes out to
A2x2 + 2ABx + B2. We've now made an expression which is the square of an integer for all integers x!
Now, we can set the coefficients of powers of x in the problem and in A2x2 + 2ABx + B2 equal to eachother.
A2 = a2
2AB = c
B2 = b2
Since a is positive, a = A, if A is positive. If A is negative, a = -A (or A = -a). The same logic leads to
b = B, if B is positive. If B is negative, b = -B (or B = -b).
Since c is positive, we know that A and B are either both positive or both negative. This leads to 2 cases:
Case 1: A is positive, B is positive. Then A = a, B = b, and
c = 2ab.
Case 2: A is negative, B is negative. Then A = -a, B = -b, and
c = 2(-a)(-b) = 2ab.
Since both cases lead to the same answer, we conclude that
c = 2ab.