Final answer:
The value of c that makes x² + 36x + c a perfect square is 324, since c must be the square of half the coefficient of the x term, resulting in (x + 18)².
Step-by-step explanation:
To determine the value of c that makes x² + 36x + c a perfect square, you must complete the square. In a perfect square trinomial a² + 2ab + b², the last term, b², is the square of half of the coefficient of the middle term. In our case, the middle term is 36x, so half of 36 is 18 and squaring it gives us 324. Therefore, c must be 324 to make x² + 36x + c a perfect square trinomial, which is (x + 18)².