Find each product. See Example 5. (4m + 2n)²

Verified step by step guidance

1. Recognize that the expression is a square of a binomial: \((4m + 2n)^2\). This means you will use the formula for the square of a sum: \((a + b)^2 = a^2 + 2ab + b^2\).
2. Identify the terms \(a\) and \(b\) in the binomial: here, \(a = 4m\) and \(b = 2n\).
3. Calculate the square of the first term: \(a^2 = (4m)^2 = 16m^2\).
4. Calculate twice the product of the two terms: \(2ab = 2 \times (4m) \times (2n) = 16mn\).
5. Calculate the square of the second term: \(b^2 = (2n)^2 = 4n^2\). Then, combine all parts to write the expanded form: \$16m^2 + 16mn + 4n^2$.

Key Concepts

Binomial Expansion

Binomial expansion involves expanding expressions raised to a power, such as (a + b)², using the formula (a + b)² = a² + 2ab + b². This allows you to rewrite the square of a sum as a sum of squares and products.

Distributive Property

The distributive property states that a(b + c) = ab + ac. It is used to multiply each term inside the parentheses by the term outside, which is essential when expanding expressions like (4m + 2n)² by treating it as (4m + 2n)(4m + 2n).

Combining Like Terms

After expanding an expression, combining like terms means adding or subtracting terms with the same variables and exponents to simplify the expression. This step is crucial to write the final product in its simplest form.