For all \(x, y \in R^{+}\) the value of \(\left[\left\\{\left(1+x+x^{2}\right)\left(1+y+y^{2}\right)\right\\} /(x y)\right]\) \(=\) (A) \(<9\) (B) \(\leq 9\) \((\mathrm{C})>9\) \((\mathrm{D}) \geq 9\)

The Arithmetic Mean-Geometric Mean Inequality, commonly known as the AM-GM Inequality, is a fundamental concept in algebra that provides a relation between the arithmetic mean and geometric mean of a set of non-negative real numbers. The inequality states that for any list of non-negative real numbers, the arithmetic mean is always greater than or equal to the geometric mean and equality holds if and only if all the numbers are equal.

The AM-GM Inequality is typically expressed as:
\[ \frac{a_1 + a_2 + \ ... + a_n}{n} \geq \sqrt[n]{a_1 \times a_2 \ \times ... \times a_n} \]
where \( a_1, a_2, ..., a_n \) are non-negative real numbers and \( n \) is the count of the numbers.

In the context of JEE Maths problems, the AM-GM Inequality can be used to derive important results and assess the minimal or maximal value of algebraic expressions, particularly when addressing inequality problems.

Simplifying algebraic expressions is a crucial skill in mathematics and is often tested in exams such as the JEE. To simplify an algebraic expression, you combine like terms, use algebraic identities, factor expressions, and cancel common factors in fractions when possible. The main goal is to make the expression easier to understand or to allow you to compare it to another expression or a specific value.

Take for example the expression \((1+x+x^2)(1+y+y^2)\) from the problem, where it is expanded and each term is carefully combined with others to avoid unnecessary complexity. By following the steps of expression expansion, common denominator creation, and terms division, it will eventually simplify the expression to a form where we can clearly apply the AM-GM Inequality and determine the nature of the inequality.

Solving inequality problems involves finding the range of values that satisfy a given inequality. Inequalities can be simple, such as linear inequalities, or more complex, involving quadratic forms or rational expressions. In the context of the JEE, inequality problems may require a mix of algebraic manipulation, conceptual understanding, and sometimes clever use of inequalities like the AM-GM.

The process often includes transforming the inequality into a simpler form, just as we simplify algebraic expressions, so that recognized inequalities can be applied. The key is always to ensure that the manipulations are valid and do not change the original inequality's solution set.

In our JEE Math problem, we applied the AM-GM Inequality after simplifying the original algebraic expression. It demonstrates how an intricate problem with variables and exponents can be reduced to a basic inequality, which leads to a clear and definite range for the expression's value. Through these methods, a student can develop strong problem-solving skills that are essential for success in examinations like the JEE and beyond.