Find each product. [(2m+7)-n]²

Verified step by step guidance

1. Recognize that the expression \( [(2m+7) - n]^2 \) is a perfect square of a binomial, which can be expanded using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \).
2. Identify \( a = (2m + 7) \) and \( b = n \) in the expression \( [(2m+7) - n]^2 \).
3. Apply the formula: \( [(2m+7) - n]^2 = (2m + 7)^2 - 2 \times (2m + 7) \times n + n^2 \).
4. Expand \( (2m + 7)^2 \) using the formula \( (x + y)^2 = x^2 + 2xy + y^2 \), where \( x = 2m \) and \( y = 7 \).
5. Write the full expanded expression by combining all terms: the square of \( 2m + 7 \), the product term \( -2(2m + 7)n \), and the square of \( n \).

Key Concepts

Binomial Expression

A binomial is an algebraic expression containing two terms connected by a plus or minus sign. In this problem, (2m + 7) - n is a binomial because it combines two terms, (2m + 7) and -n, which can be simplified or manipulated using algebraic rules.

Square of a Binomial

Squaring a binomial means multiplying the binomial by itself, such as (a - b)^2 = (a - b)(a - b). This expands to a^2 - 2ab + b^2, which is a key formula used to simplify expressions like [(2m + 7) - n]^2.

Distributive Property

The distributive property allows you to multiply each term inside a parenthesis by a term outside or in another parenthesis. It is essential for expanding products like (a - b)(a - b) by distributing each term properly to simplify the expression.