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The image contains a table called "Tabla comparativa de operaciones con funciones," which involves operations (suma, resta, multiplicación, división) applied to two functions \( f(x) \) and \( g(x) \). The functions are:

1. Suma (Addition):
   - \( f(x) = 6x^2 + 2x + 3 \)
   - \( g(x) = -3x + 5 \)
   - Notation: \( (f + g)(x) = f(x) + g(x) \)

2. Resta (Subtraction):
   - \( f(x) = 2x^2 + 2x + 3 \)
   - \( g(x) = 6x^2 + 9 \)
   - Notation: \( (f - g)(x) = f(x) - g(x) \)

3. Multiplicación (Multiplication):
   - \( f(x) = 2x - 3 \)
   - \( g(x) = -4x + 5 \)
   - Notation: \( (f \cdot g)(x) = f(x) \cdot g(x) \)

4. División (Division):
   - \( f(x) = 6x^2 + 19x + 15 \)
   - \( g(x) = 2x + 3 \)
   - Notation: \( \left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} \)

Now, let's calculate and complete the table for each operation:

### 1. **Addition (Suma):**
   \[
   (f + g)(x) = (6x^2 + 2x + 3) + (-3x + 5) = 6x^2 + (2x - 3x) + (3 + 5) = 6x^2 - x + 8
   \]
   Result: \( 6x^2 - x + 8 \)

### 2. **Subtraction (Resta):**
   \[
   (f - g)(x) = (2x^2 + 2x + 3) - (6x^2 + 9) = 2x^2 - 6x^2 + 2x - 0 + 3 - 9 = -4x^2 + 2x - 6
   \]
   Result: \( -4x^2 + 2x - 6 \)

### 3. **Multiplication (Multiplicación):**
   \[
   (f \cdot g)(x) = (2x - 3)(-4x + 5) = 2x(-4x + 5) - 3(-4x + 5) = -8x^2 + 10x + 12x - 15 = -8x^2 + 22x - 15
   \]
   Result: \( -8x^2 + 22x - 15 \)

### 4. **Division (División):**
   \[
   \left( \frac{f}{g} \right)(x) = \frac{6x^2 + 19x + 15}{2x + 3}
   \]
   This expression cannot be simplified without performing polynomial division. However, this is the result as it stands.

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Would you like more details on any of the steps or calculations?

Here are five related questions:
1. How is the polynomial division of two functions performed?
2. What are the conditions for a division of two functions to be undefined?
3. How would the results change if the functions \( f(x) \) and \( g(x) \) had higher degrees?
4. What are the rules for multiplying two binomial expressions?
5. How do you find the derivative of a sum of two functions?

**Tip:** When working with polynomial functions, it’s important to correctly group like terms to simplify expressions.

The image contains a table comparing operations on functions (addition, subtraction, multiplication, division) involving the functions f(x) and g(x).

A detailed guide on performing operations on functions, including addition, subtraction, multiplication, and division. Solve algebraic problems with functions f(x) and g(x) through polynomial operations.

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