If x^2 + xy - 32 = 0, and x and y are integers, then y could equal eac

I'm sure that it is not the easiest way but anyway: x^2+xy=32 -----> x(x+y)=32. So we have a multiplication of two integers (as x and y are integers) that yield 32. The factors of 32, that means the numbers that their multiplication by each other can yield 32, are 32, 16, 8, 4, 2, 1. Those numbers and their negative values are the only way to yield 32 by a multiplication of two integers. So in fact there is six options for the multiplication: 32x1; 16x2; 8x4; -32x-1; -16x-2; -8x-4. Now we can plug the values of y and check the answers. If the values of the two number in the multiplication matches to one of the aforementioned cases - y can be the number mentioned in the relevant option.

(A): x(x-31)=32. We can easily see that x can be equal to 32 which is a factor of 32. The multiplication is 32x1.
(B): x(x-14)=32. We can easily see that x can be equal to 16 which is a factor of 32. The multiplication is 16x2.
(C): x(x+2)=32. In this case x cannot equal to any of the factors of 32. We can simply see it by looking at the factors 32 or by using the quadratic equation that will give us an irrational number for x (which is invalid).
(D): x(x+4)=32. We can see that x can be equal to 4 which is a factor of 32. The multiplication is 8x4.
(E): x(x+14)32. We can see that x can be equal to 2 which is a factor of 32 as well. The multiplication is 16x2.

Answer: C

We can express the equation as follows:

x(x + y) = 32

Since x and y are integers, x and x + y are integer factors of 32. For example, if x = 4, then x + y = 8 which makes y = 4. So choice D is not the answer.

If x = -1, then x + y = -32 which makes y = -31. So choice A is not the answer.

If x = -2, then x + y = -16 which makes y = -14. So choice B is not the answer.

If x = 2, then x + y = 16 which makes y = 14. So choice E is not the answer.

Answer: C