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Polynomial expansion involves applying formulas or rules to expressions to simplify or redistribute them, often leading to a sum of simpler terms. A common formula used in polynomial expansion is the Binomial Theorem, which is particularly handy for expanding expressions raised to a power, like \((a + b)^2\).

This theorem states that \((a + b)^2 = a^2 + 2ab + b^2\). Each term in a binomial expansion represents a different combination of the terms in the original expression, raised to different powers.

In our exercise, substituting \(a = 2a\) and \(b = 3b\) into the formula allows us to break down the complex expression into \(4a^2 + 12ab + 9b^2\). Polynomial expansion provides a systematic method to simplify expressions and solve algebraic problems.

Combining like terms is an essential step in simplifying algebraic expressions. Like terms are terms within an expression that have the same variable parts, meaning they have identical variables raised to the same power. When combining them, you simply perform the needed arithmetic operations on their coefficients.

For example, in the expanded expression \(4a^2 + 12ab + 9b^2\), we ensure each part is already simplified in terms of like terms. Normally, combining like terms helps further reduce expressions after expansion by adding or subtracting coefficients of these terms.

This is a crucial skill because it allows for clearer final expressions and more straightforward solutions in algebraic calculations.

Algebraic expressions are combinations of numbers, symbols, and operators (like plus and minus signs) that stand in for numbers within algebra. Each expression can contain constants, variables, and operational symbols. Understanding them is key for manipulating and solving equations.

In our case, \((2a + 3b)^2\) is an algebraic expression representing a multiplication of two terms. By applying different algebraic rules like the Binomial Expansion, we can simplify it into another clearer algebraic expression: \(4a^2 + 12ab + 9b^2\).

Algebraic expressions form the foundation of higher-level math, creating pathways to solve real-world problems and complex calculations.