Expand (7x+4)(3x+4): Step-by-Step Binomial Multiplication

Q: What does FOIL stand for and how do I use it?

FOIL means First, Outer, Inner, Last. For $ (7x+4)(3x+4) $: First = $ 7x \cdot 3x $, Outer = $ 7x \cdot 4 $, Inner = $ 4 \cdot 3x $, Last = $ 4 \cdot 4 $.

Q: Why do I get four terms before combining like terms?

When you multiply two binomials, you create four separate products. Before simplifying, you should see: $ 21x^2 + 28x + 12x + 16 $. Then combine the like terms (28x + 12x = 40x).

Q: How do I know which terms are like terms?

Like terms have the exact same variable with the same exponent. In $ 21x^2 + 28x + 12x + 16 $: $ 21x^2 $ stands alone (degree 2)$ 28x $ and $ 12x $ combine (both degree 1)16 stands alone (constant)

Q: Can I use the distributive property instead of FOIL?

Absolutely! FOIL is just a special case of distribution. You can think of it as $ (7x+4) $ times each term in $ (3x+4) $, which gives the same four products.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solution

00:05 Open parentheses properly, multiply each factor by each factor

00:27 Calculate the multiplications

00:45 Collect like terms

00:53 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:

$(7x+4)(3x+4)=$

2

Step-by-step solution

Simplify the given expression, by opening the parentheses using the extended distribution law:

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

Note that in the formula template for the above distribution law, we automatically assume that the operation between the terms inside of the parentheses is addition. Furthermore remember that the sign preceding the term is an inseparable part of it. By applying the rules of sign multiplication we can present any expression inside of the parentheses. We will open the parentheses using the above formula, first as an expression where an addition operation exists between all terms. In this expression given that all terms are positive we'll proceed directly to opening the parentheses,

Let's open the parentheses:

$(\textcolor{red}{7x}+\textcolor{blue}{4})(3x+4)\\ \textcolor{red}{7x}\cdot3x+ \textcolor{red}{7x}\cdot4+\textcolor{blue}{4}\cdot 3x +\textcolor{blue}{4}\cdot4\\ 21x^2+28x+12x+16$

In calculating the above multiplications, we used the multiplication table and the laws of exponents for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

In the next step we'll combine like terms, which we will define as terms where the variable (or variables each separately) In this case x, have identical exponents (in the absence of one of the variables from the expression, we'll treat its exponent as zero power since raising any number to the zero power yields the result 1) We'll apply the commutative property of addition, furthermore we'll arrange (if needed) the expression from highest to lowest power from left to right (treating the free number as zero power):
$\textcolor{purple}{21x^2}\textcolor{green}{+28x}\textcolor{green}{+12x}+16\\ \textcolor{purple}{21x^2}\textcolor{green}{+40x}+16$

In the combining of like terms performed above, 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,

Therefore the correct answer is answer B.

3

Final Answer

$21x^2+40x+16$

Key Points to Remember

Essential concepts to master this topic

Distribution Rule: Each term multiplies every term in other binomial
FOIL Method: $7x \cdot 3x = 21x^2$ , $7x \cdot 4 = 28x$ , etc.
Check: Combine like terms: $28x + 12x = 40x$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to multiply every term by every term
Don't just multiply first terms and last terms = $21x^2 + 16$ ! This misses the middle terms completely and gives a wrong answer. Always multiply each 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

What does FOIL stand for and how do I use it?

+

FOIL means First, Outer, Inner, Last. For $(7x+4)(3x+4)$ : First = $7x \cdot 3x$ , Outer = $7x \cdot 4$ , Inner = $4 \cdot 3x$ , Last = $4 \cdot 4$ .

Why do I get four terms before combining like terms?

+

When you multiply two binomials, you create four separate products. Before simplifying, you should see: $21x^2 + 28x + 12x + 16$ . Then combine the like terms (28x + 12x = 40x).

How do I know which terms are like terms?

+

Like terms have the exact same variable with the same exponent. In $21x^2 + 28x + 12x + 16$ :

$21x^2$ stands alone (degree 2)
$28x$ and $12x$ combine (both degree 1)
16 stands alone (constant)

What if I get a different middle term when combining?

+

Double-check your multiplication! The middle terms come from $7x \cdot 4 = 28x$ and $4 \cdot 3x = 12x$ . Adding these gives $28x + 12x = 40x$ . Any other result means a calculation error.

Can I use the distributive property instead of FOIL?

+

Absolutely! FOIL is just a special case of distribution. You can think of it as $(7x+4)$ times each term in $(3x+4)$ , which gives the same four products.

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