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The problem asks to determine the remainder when \( f(x) + g(x) \) is divided by \( (x - 1)(x + 3) \). Here’s how we can approach it:

### Given functions:
1. \( f(x) = x^4 + x^3 - 2x^2 + 2x - 3 \)
2. \( g(x) = 2x^3 - 5x + 9 \)

We need to calculate the sum \( f(x) + g(x) \), then find the remainder when the result is divided by \( (x - 1)(x + 3) \).

### Step 1: Add \( f(x) \) and \( g(x) \)

Let's first add the two polynomials.

\[
f(x) + g(x) = \left( x^4 + x^3 - 2x^2 + 2x - 3 \right) + \left( 2x^3 - 5x + 9 \right)
\]

Simplifying this:

\[
f(x) + g(x) = x^4 + (1x^3 + 2x^3) - 2x^2 + (2x - 5x) + (-3 + 9)
\]
\[
f(x) + g(x) = x^4 + 3x^3 - 2x^2 - 3x + 6
\]

### Step 2: Find the remainder using the Remainder Theorem

Since we are dividing by \( (x - 1)(x + 3) \), which is a quadratic expression, we will use the fact that the remainder when dividing by a quadratic can be written as a linear polynomial:

\[
R(x) = ax + b
\]

We need to find the values of \( a \) and \( b \) such that:

\[
f(x) + g(x) = (x - 1)(x + 3) \cdot Q(x) + R(x)
\]

To do this, we can substitute \( x = 1 \) and \( x = -3 \) into the expression for \( f(x) + g(x) \) because \( (x - 1) \) and \( (x + 3) \) are factors of the divisor.

1. **Substitute \( x = 1 \) into \( f(x) + g(x) \):**
   \[
   f(1) + g(1) = 1^4 + 3(1^3) - 2(1^2) - 3(1) + 6 = 1 + 3 - 2 - 3 + 6 = 5
   \]

   So, \( R(1) = 5 \), and:
   \[
   a(1) + b = 5 \quad \Rightarrow \quad a + b = 5
   \]

2. **Substitute \( x = -3 \) into \( f(x) + g(x) \):**
   \[
   f(-3) + g(-3) = (-3)^4 + 3(-3)^3 - 2(-3)^2 - 3(-3) + 6
   \]
   \[
   = 81 + 3(-27) - 2(9) + 9 + 6 = 81 - 81 - 18 + 9 + 6 = -3
   \]

   So, \( R(-3) = -3 \), and:
   \[
   a(-3) + b = -3 \quad \Rightarrow \quad -3a + b = -3
   \]

### Step 3: Solve the system of equations

We have the following system of equations:
1. \( a + b = 5 \)
2. \( -3a + b = -3 \)

Subtract the first equation from the second:
\[
(-3a + b) - (a + b) = -3 - 5
\]
\[
-4a = -8 \quad \Rightarrow \quad a = 2
\]

Substitute \( a = 2 \) into \( a + b = 5 \):
\[
2 + b = 5 \quad \Rightarrow \quad b = 3
\]

### Step 4: Write the remainder

Thus, the remainder when \( f(x) + g(x) \) is divided by \( (x - 1)(x + 3) \) is:

\[
R(x) = 2x + 3
\]

---

Would you like more details or have any further questions?

Here are 5 related questions to deepen your understanding:

1. How do you find the quotient in polynomial division?
2. Can you explain why we use the Remainder Theorem in polynomial division?
3. What is synthetic division and how does it compare to the method used here?
4. How do we know the degree of the remainder when dividing by a quadratic expression?
5. What changes if the divisor is a cubic polynomial instead of a quadratic?

**Tip:** In problems involving polynomial division, always simplify the dividend first and look for opportunities to apply the Remainder or Factor Theorem!

Diketahui f(x) = x^4 + x^3 - 2x^2 + 2x - 3 dan g(x) = 2x^3 - 5x + 9. Jika [f(x) + g(x)] dibagi (x - 1)(x + 3), sisa pembagiannya adalah ....

Learn how to solve the polynomial division of f(x) = x^4 + x^3 - 2x^2 + 2x - 3 and g(x) = 2x^3 - 5x + 9, and find the remainder when divided by (x - 1)(x + 3). This step-by-step guide explains the addition of polynomials and the application of the Remainder Theorem. Suitable for Grades 10-12.

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