Find the value of c so that (x+1) is a factor of the polynomial p (x).

p (x) = 2x^4-5x^2+cx+1

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