Since (x+1) is a factor of the given polynomial p(x), use what we call the "Factor Theorem." In this theorem, the polynomial p(x) is equal to zero so that no remainder will exist.
Let p(x) = 0, so
2x⁴ - 5x² + cx + 1 = 0
We need to substitute a value for x. Then, use the factor x + 1 = 0 as x = -1.
2(-1)⁴ - 5(-1)² + c(-1) + 1 = 0
Note that a negative one raise to the power of a positive number will result a positive one. Solving for c,
2(1) - 5(1) + c(-1) + 1 = 0
2 - 5 - c + 1 = 0
-c = -2 + 5 - 1
-c = 2
c = -2