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To solve this, we first need to find the correct values of \(x\) that will simplify the given rational function decomposition and help us determine the coefficients \(A\), \(B\), and \(C\).

The original equation is:
\[
\frac{2x^2 + 2}{(x+2)(x^2 - 2x + 2)} = \frac{A}{x+2} + \frac{Bx + C}{x^2 - 2x + 2}.
\]

To solve for \(A\), \(B\), and \(C\), we multiply both sides by the common denominator \((x+2)(x^2 - 2x + 2)\) to obtain:
\[
2x^2 + 2 = A(x^2 - 2x + 2) + (Bx + C)(x+2).
\]

We then expand and collect like terms on the right-hand side and equate the coefficients of corresponding powers of \(x\).

### Selecting values for \(x\):
1. **Choose \(x = -2\)**: This will simplify the term \(A/(x+2)\), allowing us to solve for \(A\).
2. **Choose values that make solving easier for other coefficients \(B\) and \(C\)**. Typically, using simple numbers like \(0\) or \(1\) can simplify calculations when substituted into the expanded polynomial equation.

Thus, the useful values of \(x\) for this problem are:
- \(x = -2\)
- \(x = 0\)
- \(x = 1\)

### The correct choices are:
- \(-2\)
- \(0\)
- \(1\)

Would you like further details on how the coefficients \(A\), \(B\), and \(C\) are determined?

### Related Questions:
1. How do you expand and simplify polynomial expressions involving partial fractions?
2. What is the role of choosing specific values of \(x\) when solving for coefficients in partial fraction decomposition?
3. How do you solve systems of linear equations obtained from matching coefficients in polynomial identities?
4. What other methods can be used for partial fraction decomposition besides equating coefficients?
5. Why is partial fraction decomposition useful in integration problems?

### Tip:
When performing partial fraction decomposition, always look for values of \(x\) that will cancel terms to simplify solving for unknown coefficients.

Choose all the correct answers for the values of x that simplify the rational function decomposition to determine coefficients A, B, and C.

Learn how to determine values for x to simplify rational function decomposition and solve for coefficients A, B, and C. This problem explores partial fraction decomposition, polynomial expansion, and solving systems of equations. Ideal for students in Grades 10-12.

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