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The image you uploaded contains a list of factorization and expansion problems. Let's go through and solve each of them one by one.

### Factorization:

1. **\( x^2 = 9 \)**  
   \[
   x^2 - 9 = 0 \quad \text{(Rewrite as a difference of squares)}
   \]
   \[
   (x - 3)(x + 3) = 0
   \]
   The factorization is: **\( (x - 3)(x + 3) \)**

2. **\( x^2 - 16x \)**  
   Factor out the common term:
   \[
   x(x - 16)
   \]
   The factorization is: **\( x(x - 16) \)**

3. **\( 4x^2 - 25x \)**  
   Factor out the greatest common factor (GCF), which is \( x \):
   \[
   x(4x - 25)
   \]
   The factorization is: **\( x(4x - 25) \)**

4. **\( x^2 - 100 \)**  
   This is a difference of squares:
   \[
   (x - 10)(x + 10)
   \]
   The factorization is: **\( (x - 10)(x + 10) \)**

5. **\( x^3 - 36x \)**  
   Factor out the greatest common factor, \( x \):
   \[
   x(x^2 - 36) = x(x - 6)(x + 6)
   \]
   The factorization is: **\( x(x - 6)(x + 6) \)**

6. **\( 4x^2 + 6x \)**  
   Factor out the greatest common factor, which is \( 2x \):
   \[
   2x(2x + 3)
   \]
   The factorization is: **\( 2x(2x + 3) \)**

7. **\( x^3 + 4x \)**  
   Factor out the greatest common factor, \( x \):
   \[
   x(x^2 + 4)
   \]
   The factorization is: **\( x(x^2 + 4) \)**

8. **\( x^2 - 49 \)**  
   This is a difference of squares:
   \[
   (x - 7)(x + 7)
   \]
   The factorization is: **\( (x - 7)(x + 7) \)**

---

### Expansion:

1. **\( (x + 3)^3 \)**  
   Use the binomial expansion formula for \( (a + b)^3 \):
   \[
   (x + 3)^3 = (x + 3)(x + 3)(x + 3)
   \]
   First, expand two terms:
   \[
   (x + 3)(x + 3) = x^2 + 6x + 9
   \]
   Now, multiply the result by \( (x + 3) \):
   \[
   (x^2 + 6x + 9)(x + 3) = x^3 + 3x^2 + 6x^2 + 18x + 9x + 27
   \]
   Combine like terms:
   \[
   x^3 + 9x^2 + 27x + 27
   \]
   The expansion is: **\( x^3 + 9x^2 + 27x + 27 \)**

2. **\( (2x + 5)(2x - 5) \)**  
   This is a difference of squares:
   \[
   (2x + 5)(2x - 5) = 4x^2 - 25
   \]
   The expansion is: **\( 4x^2 - 25 \)**

---

Let me know if you need any more detailed explanations!

### Related Questions:
1. How can you recognize a difference of squares in an equation?
2. What is the general form of factoring cubic polynomials?
3. What technique is used to factor expressions like \( x^2 + bx \)?
4. How does the binomial expansion formula work for higher powers?
5. What is the relationship between factoring and solving quadratic equations?

**Tip:** Always look for a common factor to simplify expressions before attempting more complex factorization techniques!

HW: Factorise and Expand algebraic expressions.

This page covers step-by-step solutions to algebraic problems involving the factorization of quadratic and cubic expressions, including using the difference of squares and the binomial expansion formula. Suitable for students in Grades 8-10.

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