We are given the polynomial $P(x) = -2x^3 + cx^2 - 5x + 2$, and we need to find the value of $c$ such that the remainder is the same when divided by $x - 2$ and $x + 1$.

According to the Remainder Theorem, the remainder when a polynomial $P(x)$ is divided by $x - a$ is $P(a)$. Therefore, the remainder when divided by $x - 2$ is $P(2)$, and the remainder when divided by $x + 1$ is $P(-1)$.

We are told that these remainders are equal, so:

$P(2) = P(-1)$

Step 1: Calculate $P(2)$

Substitute $x = 2$ into $P(x)$:

$P(2) = -2(2)^3 + c(2)^2 - 5(2) + 2$ $P(2) = -2(8) + c(4) - 10 + 2$ $P(2) = -16 + 4c - 10 + 2$ $P(2) = 4c - 24$

Step 2: Calculate $P(-1)$

Substitute $x = -1$ into $P(x)$:

$P(-1) = -2(-1)^3 + c(-1)^2 - 5(-1) + 2$ $P(-1) = -2(-1) + c(1) + 5 + 2$ $P(-1) = 2 + c + 5 + 2$ $P(-1) = c + 9$

Step 3: Set $P(2) = P(-1)$

Now, set the two expressions equal to each other:

$4c - 24 = c + 9$

Step 4: Solve for $c$

Subtract $c$ from both sides:

$3c - 24 = 9$

Add 24 to both sides:

$3c = 33$

Finally, divide by 3:

$c = 11$

Thus, the value of $c$ that makes the remainders equal is $\boxed{11}$.

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How does the Remainder Theorem work for polynomials?
2. What is the significance of the factorization of a polynomial?
3. How would the remainder change if divided by other linear factors like $x - 3$?
4. Can synthetic division be used to solve similar problems, and how?
5. How does the value of $c$ affect the graph of the polynomial?

Tip: When working with polynomials and the Remainder Theorem, always substitute carefully to avoid errors, especially with negative signs.