To solve this, we need to analyze the rational function given and then choose appropriate values for $x$ to simplify the system and determine constants $A$, $B$, and $C$.

Problem Analysis

The function is given in a form suitable for partial fraction decomposition: $\frac{2x^2 + 2x + 3}{(x+2)^3} = \frac{A}{x+2} + \frac{B}{(x+2)^2} + \frac{C}{(x+2)^3}.$

To match both sides, we can clear the denominators by multiplying through by $(x+2)^3$, leading to an equation where the polynomial on the left equals a polynomial on the right. We then equate coefficients of like terms.

Equation System and Solution

We already have the following equations based on this comparison:

1. $A + B + C = 2$
2. $9A + 3B + C = 6$
3. $16A + 4B + C = 11$

To solve these equations, we substitute values for $x$ that simplify the system.

Choosing Values for $x$

The values of $x$ typically chosen in such setups are ones that simplify the polynomial terms and reveal values of $A$, $B$, and $C$. In this case, we look for values that make the individual terms zero or yield simple relationships among $A$, $B$, and $C$.

Let’s review each choice for its suitability:

1. $x = -2$: This would be useful as it zeros out certain terms.
2. Other values: 0, -1, -4, 3, -3, 1, and 2 are typically tested to observe their impact on simplifying the equations.

Thus, based on the standard partial fraction approach, we should select values that make substitution straightforward while ensuring they satisfy all three equations