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We are given two functions:

\[
f(x) = \frac{x - 1}{x^2 + 3x + 2}, \quad g(x) = \frac{-2x + 5}{2x^2 + 3x - 5}
\]

The goal is to find the domain of the function \((f + g)(x)\).

### Steps to determine the domain:
1. **Identify restrictions for each function**: The domain of a rational function is the set of all real numbers except those that make the denominator equal to zero.

2. **Find the values of \(x\) that make the denominator of \(f(x)\) zero**:
   \[
   x^2 + 3x + 2 = 0
   \]
   This quadratic can be factored:
   \[
   (x + 1)(x + 2) = 0
   \]
   So, \(x = -1\) and \(x = -2\) make the denominator of \(f(x)\) zero, meaning \(f(x)\) is undefined for \(x = -1\) and \(x = -2\).

3. **Find the values of \(x\) that make the denominator of \(g(x)\) zero**:
   \[
   2x^2 + 3x - 5 = 0
   \]
   We can solve this quadratic using the quadratic formula:
   \[
   x = \frac{-3 \pm \sqrt{(3)^2 - 4(2)(-5)}}{2(2)} = \frac{-3 \pm \sqrt{9 + 40}}{4} = \frac{-3 \pm \sqrt{49}}{4} = \frac{-3 \pm 7}{4}
   \]
   So the solutions are:
   \[
   x = \frac{-3 + 7}{4} = 1, \quad x = \frac{-3 - 7}{4} = -\frac{5}{2}
   \]
   Thus, \(x = 1\) and \(x = -\frac{5}{2}\) make the denominator of \(g(x)\) zero, meaning \(g(x)\) is undefined for \(x = 1\) and \(x = -\frac{5}{2}\).

### Domain of \((f + g)(x)\):
The domain of \((f + g)(x)\) is the set of all real numbers except the values that make either \(f(x)\) or \(g(x)\) undefined. These values are:
- \(x = -1\)
- \(x = -2\)
- \(x = 1\)
- \(x = -\frac{5}{2}\)

Thus, the domain of \((f + g)(x)\) is all real numbers except \(x = -1\), \(x = -2\), \(x = 1\), and \(x = -\frac{5}{2}\).

### Final answer:
The domain of \((f + g)(x)\) is:
\[
\mathbb{R} \setminus \left\{-2, -1, -\frac{5}{2}, 1\right\}
\]

Would you like further details on any step or have other questions?

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Here are 5 related questions for practice:
1. How would you find the domain of \((f - g)(x)\)?
2. What is the domain of the function \((f \times g)(x)\)?
3. How would you calculate the domain of \(\frac{f(x)}{g(x)}\)?
4. What happens to the domain if one of the numerators had a variable that could also become zero?
5. How can you graphically interpret the domain of a rational function?

**Tip**: Always check both the numerator and denominator for restrictions when determining the domain of a rational function!

Diketahui f(x) = \frac{x-1}{x^2 + 3x + 2} dan g(x) = \frac{-2x + 5}{2x^2 + 3x - 5}. Domain fungsi (f+g)(x) adalah ....

Learn how to find the domain of the sum of two rational functions: f(x) = \frac{x-1}{x^2 + 3x + 2} and g(x) = \frac{-2x + 5}{2x^2 + 3x - 5}. This problem covers important algebraic concepts, including rational functions and solving quadratic equations, suitable for high school students.

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