Find each product. See Examples 5 and 6. [(2p-3)+q]^2

Verified step by step guidance

1. Identify the expression to be squared: \((2p - 3) + q\).
2. Recognize that this is a binomial squared, which can be expanded using the formula \((a + b)^2 = a^2 + 2ab + b^2\).
3. Assign \(a = (2p - 3)\) and \(b = q\).
4. Apply the formula: \(((2p - 3) + q)^2 = ((2p - 3))^2 + 2((2p - 3))(q) + q^2\).
5. Expand each term: \((2p - 3)^2\), \(2(2p - 3)q\), and \(q^2\).

Key Concepts

Binomial Expansion

Binomial expansion is a method used to expand expressions that are raised to a power, particularly those in the form of (a + b)^n. The expansion is achieved using the Binomial Theorem, which states that (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where k ranges from 0 to n. This theorem allows for systematic calculation of each term in the expansion.

Squaring a Binomial

Squaring a binomial involves applying the formula (a + b)^2 = a^2 + 2ab + b^2. This formula simplifies the process of multiplying a binomial by itself, ensuring that all terms are accounted for. In the context of the given expression, it will help in determining the individual components of the squared expression.

Combining Like Terms

Combining like terms is a fundamental algebraic process that involves simplifying expressions by adding or subtracting terms that have the same variable raised to the same power. This step is crucial after expanding a polynomial, as it helps to condense the expression into a more manageable form, making it easier to interpret or further manipulate.