Find each product. (4m+2n)²

Verified step by step guidance

1. Recognize that the expression \( (4m + 2n)^2 \) is a binomial squared, which can be expanded using the formula for the square of a sum: \( (a + b)^2 = a^2 + 2ab + b^2 \).
2. Identify \( a = 4m \) and \( b = 2n \) in the expression \( (4m + 2n)^2 \).
3. Calculate \( a^2 \) by squaring \( 4m \), which means squaring both the coefficient and the variable: \( (4m)^2 = 4^2 \times m^2 \).
4. Calculate \( 2ab \) by multiplying 2, \( a = 4m \), and \( b = 2n \): \( 2 \times 4m \times 2n \).
5. Calculate \( b^2 \) by squaring \( 2n \): \( (2n)^2 = 2^2 \times n^2 \), then combine all parts to write the expanded form: \( a^2 + 2ab + b^2 \).

Key Concepts

Binomial Square Formula

The binomial square formula states that (a + b)^2 = a^2 + 2ab + b^2. It is used to expand the square of a sum of two terms by squaring each term and adding twice their product.

Distributive Property

The distributive property allows multiplication over addition, meaning a(b + c) = ab + ac. This property is essential when expanding expressions like (4m + 2n)^2 by distributing each term properly.

Combining Like Terms

After expanding an expression, like terms (terms with the same variable and exponent) must be combined to simplify the result. This step ensures the final expression is in its simplest form.