Verified Answer :

  • Identify the distributive property: \(a(b - c - d) = ab - ac - ad\).
  • Multiply each term inside the parentheses by the factor 2: \(2 \cdot 6e = 12e\), \(2 \cdot (-3f) = -6f\), and \(2 \cdot (-4) = -8\).
  • Combine the results into the simplified expression \(12e - 6f - 8\).
  • The final equivalent expression is \(\boxed{12e - 6f - 8}\).

Explanation

  1. Understanding the Distributive Property

The expression given is \((6e - 3f - 4) \cdot 2\). This is a simple algebraic expression involving a trinomial and a multiplier. Our objective is to simplify this expression by distributing the constant 2 to each term inside the parentheses. The distributive property of multiplication states that for any numbers \(a, b, c,\) and \(d\), the product \(a(b - c - d)\) is equal to \(ab - ac - ad\). This rule allows us to expand the expression by multiplying each term inside the parentheses by 2.

  1. Applying the Distributive Property

To distribute the factor 2 into the trinomial \((6e - 3f - 4)\), we multiply each term individually by 2:

  • First, multiply \(6e\) by 2: \(2 \cdot 6e = 12e\).
  • Second, multiply \(-3f\) by 2: \(2 \cdot (-3f) = -6f\).
  • Third, multiply \(-4\) by 2: \(2 \cdot (-4) = -8\).

Combining these parts, we get \(12e - 6f - 8\).

  1. Finalizing the Simplified Expression

By performing the multiplication, we have transformed the original expression into its expanded and simplified form: \(12e - 6f - 8\). This form contains the fewest symbols possible and correctly represents the product of the given expression. Thus, \((6e - 3f - 4) \cdot 2 = 12e - 6f - 8\).

Examples

The distributive property is a fundamental tool used in many real-life applications. For example, if you are buying 2 sets of school supplies, each containing 6 erasers (e), 3 folders (f), and 4 pencils, the total number of items can be expressed as \(2 \cdot (6e - 3f - 4)\). Distributing the 2 helps you quickly calculate the total amount of each supply you will have. Mastering this concept makes it much easier to manage inventory or calculate costs in everyday situations.

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