Find the continued product:
(i) `(x + 1)(x - 1) (x^(2) + 1)`
(ii) `(x - 3)(x + 3)(x^(2) + 9)`
(iii) `(3x - 2y)(3x + 2y)(9x^(2) + 4y^(2))`
(iv) `(2p + 3)(2p - 3)(4p^(2) + 9)`

Answer

To find the continued product for each part, we will use the formula for the difference of squares, which states that:

\[ (A + B)(A - B) = A^2 - B^2 \]

We will apply this formula step by step for each part of the question.

### Part (i): \((x + 1)(x - 1)(x^2 + 1)\)

1. **Identify the first two factors**:
\((x + 1)(x - 1)\) can be simplified using the difference of squares formula.
\[ (x + 1)(x - 1) = x^2 - 1 \]

2. **Combine with the third factor**:
Now we multiply this result with the third factor:
\[ (x^2 - 1)(x^2 + 1) \]

3. **Apply the difference of squares again**:
\[ (x^2 - 1)(x^2 + 1) = (x^2)^2 - (1)^2 = x^4 - 1 \]

**Final Answer for Part (i)**:
\[ x^4 - 1 \]

### Part (ii): \((x - 3)(x + 3)(x^2 + 9)\)

1. **Identify the first two factors**:
\((x - 3)(x + 3)\) can be simplified using the difference of squares formula.
\[ (x - 3)(x + 3) = x^2 - 9 \]

2. **Combine with the third factor**:
Now we multiply this result with the third factor:
\[ (x^2 - 9)(x^2 + 9) \]

3. **Apply the difference of squares again**:
\[ (x^2 - 9)(x^2 + 9) = (x^2)^2 - (9)^2 = x^4 - 81 \]

**Final Answer for Part (ii)**:
\[ x^4 - 81 \]

### Part (iii): \((3x - 2y)(3x + 2y)(9x^2 + 4y^2)\)

1. **Identify the first two factors**:
\((3x - 2y)(3x + 2y)\) can be simplified using the difference of squares formula.
\[ (3x - 2y)(3x + 2y) = (3x)^2 - (2y)^2 = 9x^2 - 4y^2 \]

2. **Combine with the third factor**:
Now we multiply this result with the third factor:
\[ (9x^2 - 4y^2)(9x^2 + 4y^2) \]

3. **Apply the difference of squares again**:
\[ (9x^2 - 4y^2)(9x^2 + 4y^2) = (9x^2)^2 - (4y^2)^2 = 81x^4 - 16y^4 \]

**Final Answer for Part (iii)**:
\[ 81x^4 - 16y^4 \]

### Part (iv): \((2p + 3)(2p - 3)(4p^2 + 9)\)

1. **Identify the first two factors**:
\((2p + 3)(2p - 3)\) can be simplified using the difference of squares formula.
\[ (2p + 3)(2p - 3) = (2p)^2 - (3)^2 = 4p^2 - 9 \]

2. **Combine with the third factor**:
Now we multiply this result with the third factor:
\[ (4p^2 - 9)(4p^2 + 9) \]

3. **Apply the difference of squares again**:
\[ (4p^2 - 9)(4p^2 + 9) = (4p^2)^2 - (9)^2 = 16p^4 - 81 \]

**Final Answer for Part (iv)**:
\[ 16p^4 - 81 \]

### Summary of Answers:
1. Part (i): \(x^4 - 1\)
2. Part (ii): \(x^4 - 81\)
3. Part (iii): \(81x^4 - 16y^4\)
4. Part (iv): \(16p^4 - 81\)