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resuelve los siguientes ejercicios correctamente

Let's start by solving the exercises from the images you provided. I'll go through each one step by step.

### First Image: ADA 2 - Productos notables y factorización

1. **Desarrollar:**
   - \((x + 2x^2 - 3x)(x - 3)\)
   
   Let's expand this expression:
   \[
   (x + 2x^2 - 3x)(x - 3) = x \cdot (x - 3) + 2x^2 \cdot (x - 3) - 3x \cdot (x - 3)
   \]
   Simplifying:
   \[
   = x^2 - 3x + 2x^3 - 6x^2 - 3x^2 + 9x
   \]
   \[
   = 2x^3 - 9x^2 + 6x
   \]

2. **Factorizar:**
   - \(2x^2 + 4x - 6\)

   Factoring out the greatest common factor:
   \[
   = 2(x^2 + 2x - 3)
   \]
   Factoring the quadratic:
   \[
   = 2(x - 1)(x + 3)
   \]

3. **Hallar \(A \times (B + C - D)\):**
   - Given:
     - \(A = (x + 10)(x - 4)\)
     - \(B = 3x + 1\)
     - \(C = 3x^2 - 6x\)
     - \(D = 10x\)
   
   Simplifying:
   \[
   A = x^2 - 4x + 10x - 40 = x^2 + 6x - 40
   \]
   \[
   B + C - D = (3x + 1) + (3x^2 - 6x) - 10x = 3x^2 - 13x + 1
   \]
   Then multiply \(A \times (B + C - D)\):
   \[
   (x^2 + 6x - 40) \times (3x^2 - 13x + 1)
   \]
   The detailed expansion would be a long polynomial.

4. **Factorizar:**
   - \(x^3 + 1\)
   
   This is a sum of cubes:
   \[
   x^3 + 1 = (x + 1)(x^2 - x + 1)
   \]

5. **Hallar \(x + y^2\):**
   - If \(x = 3\) and \(y = 2\):
   \[
   x + y^2 = 3 + 2^2 = 3 + 4 = 7
   \]

6. **Resolver:**
   - \( (5 - x) + 3 = x + 1 \)
   
   Solving for \(x\):
   \[
   5 - x + 3 = x + 1
   \]
   \[
   8 - x = x + 1
   \]
   \[
   8 - 1 = 2x
   \]
   \[
   x = \frac{7}{2} = 3.5
   \]

7. **Hallar \(A + B + C\):**
   - Given:
     - \(A = 4x\)
     - \(B = x^2\)
     - \(C = x + 7\)
   
   Summing:
   \[
   A + B + C = 4x + x^2 + x + 7 = x^2 + 5x + 7
   \]

### Second Image: ADA 1 - Operaciones básicas con números reales

8. **Problemas de aplicación:**
   - Given problem about a box with 3 candies per side:
   
   Let's set up the equation with the given information. The problem states that the box fits \(3\) candies per side; if each side length increases, \(x\):
   
   \[
   \text{Volume of a box} = \text{side length}^3 = x^3
   \]

9. **Problema de aplicación:**
   - A problem involving buying \(11.99\) meters of ribbon.
   
   This is more of a real-world application; if given the price per meter and a total length, you would calculate the total cost using multiplication.

10. **Problema extra:**
    - Factor \(x^5 + 3x^4 - x^3 - 3x^2 + x + 3\).

    Factoring step by step:
    - Check for common factors and then apply factoring techniques (grouping, synthetic division).

If you would like more details or need help with specific problems from this set, let me know!

### Related Questions:
1. What are other methods to factor polynomials?
2. How do you solve real-world problems involving algebraic expressions?
3. How can the distributive property be used to simplify expressions?
4. What are some tips for recognizing patterns in polynomial factorization?
5. How can you check if a factorization is correct?

**Tip:** Always double-check your factorization by expanding the factors back into the original polynomial to verify your work!

Aprende a resolver correctamente los ejercicios de productos notables, factorización de polinomios y operaciones básicas con números reales con una solución detallada paso a paso. Estos problemas cubren conceptos clave de álgebra, incluyendo productos notables, factorización de polinomios y operaciones básicas con números reales. Adecuado para estudiantes de grados 7-9.

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