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The problem you provided contains three parts involving the function \( f(x) = (a + 1)x^2 - (a + 3)x + a - 2 \). Let's analyze and solve each part step by step.

### Part (I)
The first part asks us to find the value of the real number \( a \) such that the graph of the function \( f(x) \) is symmetric with respect to the line \( x = 1 \).

To find the value of \( a \) for which the parabola is symmetric about \( x = 1 \), we set the vertex of the parabola to be at \( x = 1 \). The vertex of a quadratic function \( ax^2 + bx + c \) is located at \( x = -\frac{b}{2a} \).

For the given function:
\[
f(x) = (a + 1)x^2 - (a + 3)x + a - 2
\]
The vertex is at:
\[
x = -\frac{-(a + 3)}{2(a + 1)} = \frac{a + 3}{2(a + 1)}
\]

Setting the vertex to \( x = 1 \):
\[
\frac{a + 3}{2(a + 1)} = 1
\]
Solving this equation will give us the value of \( a \).

### Part (II)
The second part asks us to solve the inequality \( f(x) \leq \frac{-2x^2 + 3x + 7}{x + 1} \) for \( a = -1 \).

Plug in \( a = -1 \) into the expression for \( f(x) \) and simplify the inequality. Then, solve the resulting inequality step by step.

### Part (III)
The third part requires us to find the range of values for \( a \) such that the inequality \( f(x) \geq x^2 - 2x - 2 \) holds for all \( x \in (0, +\infty) \).

To solve this, we can use the method of extreme values by finding the minimum or maximum of the difference between \( f(x) \) and the given quadratic expression over the interval and ensuring it meets the condition.

Would you like me to proceed with the detailed step-by-step calculations for each part? 

### Related Questions
1. What are the properties of a parabola that determine its symmetry with respect to a vertical line?
2. How do you find the vertex of a quadratic equation, and why is it important for symmetry?
3. What methods can be used to solve quadratic inequalities?
4. How does changing the parameter \( a \) affect the shape and position of the graph of \( f(x) \)?
5. What role do extreme values play in ensuring that inequalities hold true for all values in a given range?

**Tip:** When dealing with quadratic functions, understanding the vertex and its position is key to analyzing the symmetry and solving related problems.

已知函数 f(x) = (a + 1)x^2 - (a + 3)x + a - 2. 求解关于 a 的方程和不等式问题.

This problem involves solving a quadratic function for symmetry about x = 1 and solving inequalities for a given parameter a. Learn step-by-step how to determine the value of a and solve quadratic inequalities.

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