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The distributive property is a fundamental concept in algebra. It allows you to multiply a single term by each term within a set of parentheses. This property is expressed as: \(a(b + c) = ab + ac\).
In applying this rule with our given exercise, we use the distributive property to multiply the monomial \(4\) by each term inside the binomial \(2a + 6b\).
By distributing the \(4\), each term inside the parentheses is multiplied individually, making complex algebraic equations simpler to handle.

A monomial is a single-term expression consisting of a constant, a variable, or a product of both. Examples include \(7\), \(3x\), and \(4a^2b\).
In our exercise, the monomial is \(4\), which means it has only one term.
When you multiply a monomial by each term within a binomial, the distributive property comes into play. This multiplication knocks down the complexity step by step.

A binomial is an algebraic expression containing exactly two different terms separated by a plus (+) or minus (−) sign. Examples are \(x+y\) and \(3a-4b\).
For the exercise we’re considering, \(2a+6b\) is the binomial.
Each term in the binomial is treated separately while applying the distributive property, breaking it into smaller parts to simplify calculation.

Let's dive deeply into the exercise we've worked on. Here are the steps we followed:
1. **Understand the Problem**
Recognize our task as multiplying a monomial by a binomial. The given expression is \(4(2a+6b)\).
2. **Apply the Distributive Property**
We use the rule \(a(b+c) = ab + ac\) to distribute \(4\) to each term inside the parentheses.
3. **Multiply 4 by Each Term**
Start with the first term: \(4 \times 2a = 8a\). Then, proceed with the second term: \(4 \times 6b = 24b\).
4. **Combine the Results**
Adding the individual results, we get the final expression: \(8a + 24b\).
This straightforward, systematic approach ensures that you understand how to handle multiplying monomials with binomials.