Find each of the following products:
(i) (4x + 5y) (4x - 5y) (ii) `(3x^(2) + 2y^(2)) (3x^(2) - 2y^(2))`

Answer

Let's solve the given problems step by step.

### Problem (i): Find the product of (4x + 5y)(4x - 5y)

1. **Identify the expression**: We have two binomials, (4x + 5y) and (4x - 5y).
2. **Recognize the pattern**: This expression can be recognized as a difference of squares, which follows the formula:
\[ (a + b)(a - b) = a^2 - b^2 \]
Here, \( a = 4x \) and \( b = 5y \).
3. **Apply the formula**:
\[ (4x + 5y)(4x - 5y) = (4x)^2 - (5y)^2 \]
4. **Calculate \( a^2 \) and \( b^2 \)**:
\[ (4x)^2 = 16x^2 \quad \text{and} \quad (5y)^2 = 25y^2 \]
5. **Substitute back into the equation**:
\[ 16x^2 - 25y^2 \]

Thus, the product of (4x + 5y)(4x - 5y) is:
\[ \boxed{16x^2 - 25y^2} \]

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### Problem (ii): Find the product of (3x² + 2y²)(3x² - 2y²)

1. **Identify the expression**: We have two binomials, (3x² + 2y²) and (3x² - 2y²).
2. **Recognize the pattern**: This expression can also be recognized as a difference of squares:
\[ (a + b)(a - b) = a^2 - b^2 \]
Here, \( a = 3x^2 \) and \( b = 2y^2 \).
3. **Apply the formula**:
\[ (3x^2 + 2y^2)(3x^2 - 2y^2) = (3x^2)^2 - (2y^2)^2 \]
4. **Calculate \( a^2 \) and \( b^2 \)**:
\[ (3x^2)^2 = 9x^4 \quad \text{and} \quad (2y^2)^2 = 4y^4 \]
5. **Substitute back into the equation**:
\[ 9x^4 - 4y^4 \]

Thus, the product of (3x² + 2y²)(3x² - 2y²) is:
\[ \boxed{9x^4 - 4y^4} \]

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