Unlocking Algebraic Expressions: How to Multiply Out and Simplify \(4(x + 3) + m(x + 3)\)

Simplifying algebraic expressions is a fundamental skill in mathematics. It involves rewriting an expression in its most compact and clear form. Let's explore how to "multiply out of the bracket and simplify" the expression \(4(x + 3) + m(x + 3)\).

Key Insights

The Distributive Property is Fundamental: This property, \(a(b + c) = ab + ac\), is the primary tool for multiplying out terms from brackets.
Combining Like Terms is Crucial for Simplification: After expanding, terms with the same variable raised to the same power (or constant terms) are grouped together.
Multiple Paths to Simplification: The expression can be simplified by first distributing each multiplier or by first factoring out a common binomial term. Both methods yield equivalent results.

The Cornerstone: Understanding the Distributive Property

The distributive property is a key algebraic concept that allows us to multiply a single term by a sum or difference of terms enclosed in parentheses. In its general form, it's expressed as:

\(a(b + c) = ab + ac\)

This means that the term 'a' outside the parenthesis is distributed (multiplied) to each term ('b' and 'c') inside the parenthesis. This principle is essential for expanding bracketed expressions before simplification.

Mathematical formulas on a chalkboard illustrating algebraic concepts

Algebraic expressions often involve applying fundamental properties like distributivity.

Step-by-Step Simplification: Multiplying Out and Combining

Let's break down the expression \(4(x + 3) + m(x + 3)\) step by step.

Step 1: Apply the Distributive Property to Each Term

We have two terms involving parentheses: \(4(x + 3)\) and \(m(x + 3)\). We'll apply the distributive property to each one.

Expanding the First Term: \(4(x + 3)\)

Multiply 4 by each term inside the first parenthesis:

\(4 \times x = 4x\)

\(4 \times 3 = 12\)

So, \(4(x + 3) = 4x + 12\).

Expanding the Second Term: \(m(x + 3)\)

Similarly, multiply \(m\) by each term inside the second parenthesis:

\(m \times x = mx\)

\(m \times 3 = 3m\)

So, \(m(x + 3) = mx + 3m\).

Step 2: Combine the Expanded Terms

Now, substitute these expanded forms back into the original expression:

\(4(x + 3) + m(x + 3) = (4x + 12) + (mx + 3m)\)

Since we are adding, we can remove the parentheses:

\(4x + 12 + mx + 3m\)

Step 3: Group and Simplify Like Terms

"Like terms" are terms that contain the same variable raised to the same power. Constants are also like terms with each other.

Terms with \(x\): We have \(4x\) and \(mx\). These can be combined by factoring out \(x\):
\(4x + mx = (4 + m)x\)
Constant and \(m\) terms: We have \(12\) (a constant) and \(3m\) (a term with \(m\)). These are not like terms with each other (unless \(m\) is a specific number that makes \(3m\) a constant that can be combined with 12, but \(m\) is treated as a variable here). So, they remain as \(12 + 3m\).

Putting it all together, the simplified expression is:

\((4 + m)x + 12 + 3m\)

This can also be written as \((m + 4)x + (3m + 12)\) for conventional ordering.

Alternative Approach: Factoring Out the Common Binomial

Another way to approach this problem is to notice that both parts of the expression, \(4(x + 3)\) and \(m(x + 3)\), share a common factor of \((x + 3)\). We can factor this common binomial out:

\(4(x + 3) + m(x + 3)\)

Treat \((x + 3)\) as a single entity. If we let \(Y = (x + 3)\), the expression becomes \(4Y + mY\).

Factoring out \(Y\), we get:

\(Y(4 + m)\)

Now, substitute back \(Y = (x + 3)\):

\((x + 3)(4 + m)\)

Or, more conventionally written: \((4 + m)(x + 3)\).

If you expand this factored form using the distributive property (or FOIL method for binomials), you'll arrive at the same result as before:

\((4 + m)(x + 3) = 4(x + 3) + m(x + 3) = 4x + 4(3) + mx + m(3) = 4x + 12 + mx + 3m\)

This confirms both approaches lead to equivalent simplified forms.

Visualizing the Simplification Path

This mindmap illustrates the two main pathways to simplify the expression \(4(x + 3) + m(x + 3)\). It shows the steps involved in both distributing first and factoring first, leading to the equivalent simplified forms.

mindmap root["Simplify: \(4(x + 3) + m(x + 3)\)"] Method1["Method 1: Distribute then Combine"] Step1_1["Apply Distributive Property"] Term1["Distribute 4:
\(4(x + 3) \Rightarrow 4x + 12\)"] Term2["Distribute m:
\(m(x + 3) \Rightarrow mx + 3m\)"] Step1_2["Combine Expanded Terms"] Expr1["Sum: \(4x + 12 + mx + 3m\)"] Step1_3["Group Like Terms"] XTerms["Terms with x:
\(4x + mx \Rightarrow (4 + m)x\)"] Const_M_Terms["Constant and m terms:
\(12 + 3m\)"] Result1["Simplified Form 1:
\((4 + m)x + 12 + 3m\)"] Method2["Method 2: Factor out Common Term \((x+3)\)"] Step2_1["Identify Common Factor"] CommonFactor["\((x+3)\) is common to both \(4(x+3)\) and \(m(x+3)\)"] Step2_2["Factor out \((x+3)\)"] Expr2["\( (x+3)(4+m) \)"] Result2["Simplified Form 2:
\((4 + m)(x + 3)\)"] Equivalence["Equivalence Check"] ExpandR2["Expanding \((4+m)(x+3)\) gives:
\(4x + 12 + mx + 3m\),
matching Form 1."]

Comparing Simplification Strategies

The choice between distributing first or factoring first can depend on the complexity of the expression and personal preference. The radar chart below offers a conceptual comparison of these two strategies based on several aspects of the simplification process for an expression like the one given.

This chart suggests that "Distributing then Combining" is very clear in its application of the distributive property, while "Factoring First" can be more efficient and offer greater step reduction if the common factor is easily identified. Both require a good conceptual understanding and care to avoid errors.

Summary of Expansion and Simplification

The following table summarizes the key transformations in the process of simplifying \(4(x + 3) + m(x + 3)\) by first multiplying out the brackets:

Original Component	After Distributive Property
\(4(x + 3)\)	\(4x + 12\)
\(m(x + 3)\)	\(mx + 3m\)
Full Expression (Expanded)	\(4x + 12 + mx + 3m\)
Full Expression (Simplified by Grouping Like Terms)	\((4 + m)x + 12 + 3m\)
Full Expression (Simplified by Factoring Common Binomial)	\((4 + m)(x + 3)\)

Visual Learning: The Distributive Property in Action

Understanding the distributive property is key to expanding expressions. This video provides a clear explanation and examples of how the distributive property works in algebra, which is directly applicable to the expression we've simplified.

The video "Algebra Basics: The Distributive Property - Math Antics" breaks down the concept \(a(b + c) = ab + ac\), showing how a term outside parentheses multiplies each term inside. This is precisely what we did for \(4(x+3)\) to get \(4x+12\), and for \(m(x+3)\) to get \(mx+3m\).