Find the product :
`(i) (x+3)(x^(2)-3x+9) " " (ii) (7+5b)(49-35b+25b^(2))`
`(iii) (5a+(1)/(2)) (25a^(2)-(5a)/(2)+(1)/(4)) " " (iv) (a+b-2)[a^(2)+b^(2)+2ab+2(a+b)+4]`.

Answer

Let's solve the given products step by step.

### (i) Find the product: \((x+3)(x^2-3x+9)\)

1. **Identify the terms**: We have \(A = x\) and \(B = 3\).
2. **Recognize the identity**: This expression resembles the identity for \(A^3 + B^3\), where:
\[ A^3 + B^3 = (A + B)(A^2 - AB + B^2) \]
3. **Calculate \(A^3\) and \(B^3\)**:
- \(A^3 = x^3\)
- \(B^3 = 3^3 = 27\)
4. **Calculate \(AB\)**:
- \(AB = x \cdot 3 = 3x\)
5. **Substitute into the identity**:
\[ (x + 3)(x^2 - 3x + 9) = x^3 + 27 \]

**Final Answer for (i)**: \(x^3 + 27\)

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### (ii) Find the product: \((7 + 5b)(49 - 35b + 25b^2)\)

1. **Identify the terms**: Here, \(A = 7\) and \(B = 5b\).
2. **Recognize the identity**: This also resembles \(A^3 + B^3\):
\[ A^3 + B^3 = (A + B)(A^2 - AB + B^2) \]
3. **Calculate \(A^3\) and \(B^3\)**:
- \(A^3 = 7^3 = 343\)
- \(B^3 = (5b)^3 = 125b^3\)
4. **Calculate \(AB\)**:
- \(AB = 7 \cdot 5b = 35b\)
5. **Substitute into the identity**:
\[ (7 + 5b)(49 - 35b + 25b^2) = 343 + 125b^3 \]

**Final Answer for (ii)**: \(343 + 125b^3\)

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### (iii) Find the product: \(\left(5a + \frac{1}{2}\right)\left(25a^2 - \frac{5a}{2} + \frac{1}{4}\right)\)

1. **Identify the terms**: Here, \(A = 5a\) and \(B = \frac{1}{2}\).
2. **Recognize the identity**: This expression resembles \(A^3 + B^3\):
\[ A^3 + B^3 = (A + B)(A^2 - AB + B^2) \]
3. **Calculate \(A^3\) and \(B^3\)**:
- \(A^3 = (5a)^3 = 125a^3\)
- \(B^3 = \left(\frac{1}{2}\right)^3 = \frac{1}{8}\)
4. **Calculate \(AB\)**:
- \(AB = 5a \cdot \frac{1}{2} = \frac{5a}{2}\)
5. **Substitute into the identity**:
\[ \left(5a + \frac{1}{2}\right)\left(25a^2 - \frac{5a}{2} + \frac{1}{4}\right) = 125a^3 + \frac{1}{8} \]

**Final Answer for (iii)**: \(125a^3 + \frac{1}{8}\)

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### (iv) Find the product: \((a + b - 2)(a^2 + b^2 + 2ab + 2(a + b) + 4)\)

1. **Identify the terms**: Here, let \(A = a + b\) and \(B = 2\).
2. **Recognize the identity**: This resembles \(A^3 - B^3\):
\[ A^3 - B^3 = (A - B)(A^2 + AB + B^2) \]
3. **Calculate \(A^3\) and \(B^3\)**:
- \(A^3 = (a + b)^3\)
- \(B^3 = 2^3 = 8\)
4. **Substitute into the identity**:
\[ (a + b - 2)((a + b)^2 + 2(a + b) + 4) = (a + b)^3 - 8 \]

**Final Answer for (iv)**: \((a + b)^3 - 8\)

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