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When working with algebraic expressions, especially binomials, the FOIL method offers a straightforward approach to multiplication. FOIL is an acronym that stands for First, Outer, Inner, Last, which represents the order of multiplying the terms within two binomials.

For example, consider the expression \( (2c - 7)^2 \), which can be rewritten as \( (2c - 7)(2c - 7) \). Using the FOIL method, we multiply the First terms of each binomial (\(2c * 2c\)), the Outer terms (\(2c * -7\)), the Inner terms (\( -7 * 2c\)), and finally the Last terms (\( -7 * -7\)). This method ensures that each term in the first binomial is multiplied by each term in the second binomial, giving us the expanded expression before combining like terms.

The distributive property is a foundation in algebra that allows us to simplify expressions by distributing a factor over terms within parentheses. This key property applies to both addition and multiplication. In essence, it lets us multiply a single term across each term within a set of parentheses.

Considering our example \( (2c - 7)^2 \), after applying the FOIL method, the distributive property is what actually allows us to say \(2c \times 2c + 2c \times -7 + -7 \times 2c + -7 \times -7 \). In other words, each term outside the parentheses (which both happen to be \(2c - 7 \) in this case) is distributed across each term inside the parentheses, ensuring all parts of the binomial are accounted for in the expansion.

After expanding an expression, we are often left with terms that have the same variable to the same power, known as like terms. These can be simplified by combining them to create a more concise expression.

In the binomial expansion of \( (2c - 7)^2 \), after applying the FOIL method, we find that we have two instances of the term \( -14c \). These are like terms because they share the same variable, \(c\), raised to the same power (1 in this case, although it is not explicitly shown). By adding these two terms together, we get \( -28c \). This simplification is a crucial final step that leads us to the fully expanded and simplified form of the expression: \(4c^2 - 28c + 49\). The process of combining like terms is essential for achieving the most simplified form of an expression.