Expand the Binomial Product: (2x+3y)(2z+12m) Step by Step

Q: Why do I need to multiply ALL terms together?

The distributive property requires each term in the first parentheses to multiply with every term in the second parentheses. This gives you $ 2 \times 2 = 4 $ total products!

Q: How do I keep track of all the multiplications?

Use FOIL: First terms, Outer terms, Inner terms, Last terms. For $ (2x+3y)(2z+12m) $:First: $ 2x \cdot 2z = 4xz $Outer: $ 2x \cdot 12m = 24xm $Inner: $ 3y \cdot 2z = 6yz $Last: $ 3y \cdot 12m = 36ym $

Q: Can I combine any of these terms?

No! Each term has different variable combinations: xz, xm, yz, and ym. Since no two terms have identical variables, none can be combined.

Q: How do I check my answer?

Count your terms! You should have exactly 4 terms after expanding. Also verify each coefficient: $ 2 \times 2 = 4 $, $ 2 \times 12 = 24 $, $ 3 \times 2 = 6 $, $ 3 \times 12 = 36 $.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 Open parentheses properly, multiply each factor by each factor

00:25 Calculate the products

00:51 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Solve the following problem:

$(2x+3y)(2z+12m)=$

2

Step-by-step solution

Let's simplify the given expression, using the extended distribution law to open the parentheses :

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$

Note that in the formula template for the above distribution law, we take by default that the operation between the terms inside the parentheses is addition. Remember that the sign preceding the term is an inseparable part of it. By applying the laws of sign multiplication we can present any expression inside of the parentheses. We open the parentheses using the above formula, first as an expression where an addition operation exists between all terms. In this expression it's clear that all terms have a plus sign prefix. Therefore we directly proceed to opening the parentheses,

Start by opening the parentheses:

$(\textcolor{red}{2x}+\textcolor{blue}{3y})(2z+12m)\\ \textcolor{red}{2x}\cdot 2z+\textcolor{red}{2x}\cdot12m+\textcolor{blue}{3y}\cdot 2z+\textcolor{blue}{3y} \cdot12m\\ 4xz+24xm+6yz+36ym$

In the next step we'll combine like terms, which we define as terms where the variable (or variables, each separately), in this case z,m x and y, have identical exponents. (In the absence of one of the variables from the expression, we'll consider its exponent as zero power, this is due to the fact that any number raised to the power of zero equals 1),

Note that in the expression that we obtained in the last step there are four different terms. This is due to the fact that there isn't even one pair of terms where the (different) variables have the same exponent. Therefore the expression that we already obtained is the final and most simplified expression:
$\textcolor{purple}{ 4xz}\textcolor{green}{+24xm}\textcolor{black}{+6yz}\textcolor{orange}{+36ym }\\$ We highlighted the different terms using colors, and as emphasized before, we made sure that the sign preceding the term is an inseparable part of it,

We can conclude that the correct answer is answer D.

3

Final Answer

$4xz+24xm+6yz+36ym$

Key Points to Remember

Essential concepts to master this topic

FOIL Method: First, Outer, Inner, Last for systematic term multiplication
Technique: $2x \cdot 2z = 4xz$ and $3y \cdot 12m = 36ym$
Check: Count four terms total: $4xz + 24xm + 6yz + 36ym$ ✓

Common Mistakes

Avoid these frequent errors

Multiplying only matching variables together
Don't multiply just 2x with 2z and 3y with 12m = missing half the terms! This ignores cross-multiplication and gives an incomplete answer. Always multiply every term in the first binomial by every term in the second binomial.

Practice Quiz

Test your knowledge with interactive questions

$(3+20)\times(12+4)=$

FAQ

Everything you need to know about this question

Why do I need to multiply ALL terms together?

+

The distributive property requires each term in the first parentheses to multiply with every term in the second parentheses. This gives you $2 \times 2 = 4$ total products!

How do I keep track of all the multiplications?

+

Use FOIL: First terms, Outer terms, Inner terms, Last terms. For $(2x+3y)(2z+12m)$ :

First: $2x \cdot 2z = 4xz$
Outer: $2x \cdot 12m = 24xm$
Inner: $3y \cdot 2z = 6yz$
Last: $3y \cdot 12m = 36ym$

Can I combine any of these terms?

+

No! Each term has different variable combinations: xz, xm, yz, and ym. Since no two terms have identical variables, none can be combined.

What if I get confused with the signs?

+

Since all terms are positive in this problem, all results stay positive. Just focus on multiplying coefficients (numbers) and combining variables (letters) separately.

How do I check my answer?

+

Count your terms! You should have exactly 4 terms after expanding. Also verify each coefficient: $2 \times 2 = 4$ , $2 \times 12 = 24$ , $3 \times 2 = 6$ , $3 \times 12 = 36$ .

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Algebraic Technique questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

Expand the Binomial Product: (2x+3y)(2z+12m) Step by Step

Binomial Distribution with Multi-Variable Terms

❤️ Continue Your Math Journey!