Use direct method to evaluate the following product :
`(i) (x+8)(x+3) " " (ii) (x-8)(x+2) " " (iii) (x-3)(x-5)`

Answer

Let's evaluate the products step by step using the direct method.

### (i) Evaluate (x + 8)(x + 3)

1. **Multiply the first terms:**
\( x \cdot x = x^2 \)

2. **Multiply the outer terms:**
\( x \cdot 3 = 3x \)

3. **Multiply the inner terms:**
\( 8 \cdot x = 8x \)

4. **Multiply the last terms:**
\( 8 \cdot 3 = 24 \)

5. **Combine all the terms:**
\( x^2 + 3x + 8x + 24 \)

6. **Combine like terms:**
\( x^2 + (3x + 8x) + 24 = x^2 + 11x + 24 \)

**Final Answer for (i):**
\( x^2 + 11x + 24 \)

---

### (ii) Evaluate (x - 8)(x + 2)

1. **Multiply the first terms:**
\( x \cdot x = x^2 \)

2. **Multiply the outer terms:**
\( x \cdot 2 = 2x \)

3. **Multiply the inner terms:**
\( -8 \cdot x = -8x \)

4. **Multiply the last terms:**
\( -8 \cdot 2 = -16 \)

5. **Combine all the terms:**
\( x^2 + 2x - 8x - 16 \)

6. **Combine like terms:**
\( x^2 + (2x - 8x) - 16 = x^2 - 6x - 16 \)

**Final Answer for (ii):**
\( x^2 - 6x - 16 \)

---

### (iii) Evaluate (x - 3)(x - 5)

1. **Multiply the first terms:**
\( x \cdot x = x^2 \)

2. **Multiply the outer terms:**
\( x \cdot -5 = -5x \)

3. **Multiply the inner terms:**
\( -3 \cdot x = -3x \)

4. **Multiply the last terms:**
\( -3 \cdot -5 = 15 \)

5. **Combine all the terms:**
\( x^2 - 5x - 3x + 15 \)

6. **Combine like terms:**
\( x^2 + (-5x - 3x) + 15 = x^2 - 8x + 15 \)

**Final Answer for (iii):**
\( x^2 - 8x + 15 \)

---

### Summary of Answers:
1. (i) \( x^2 + 11x + 24 \)
2. (ii) \( x^2 - 6x - 16 \)
3. (iii) \( x^2 - 8x + 15 \)

---