let 'x' and 'y' represent the integers

One integer is 18 more than twice another:

y = 2x + 18

their product is equal to -36:

x*y = -36

plug y = 2x + 18 into the last equation:

x*(2x + 18) = -36

by solving we find:

x1 = -3

x2 = -6

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for x = -3 we have

y = 2*(-3) + 18

y = 12

for x = -6 we have

y = 2*(-6) + 18

y = 6

we found 2 solutions:

first solution: the integers are -3 and 12

second solution: the integers are -6 and 6.