What is the <strong>MINIMUM</strong> value of (\frac{a^{2}+3a+1}{a})(\frac{b^{2}+3b+1}{b}) for a, b \gt 0 ?

Explanation

The given algebraic expression can be split into two separate fractional parts: (a + 3 + \frac{1}{a})(b + 3 + \frac{1}{b}). By the Arithmetic Mean-Geometric Mean (AM-GM) inequality, the sum of any positive real number and its reciprocal, such as x + \frac{1}{x}, is always greater than or equal to 2. Substituting this minimum value of 2 for both (a + \frac{1}{a}) and (b + \frac{1}{b}), the expression evaluates to (2 + 3)(2 + 3) = 5 \times 5 = 25.

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