Match Quadratic Expressions: (x+4)(x-3) and Their Expanded Forms

Q: What does FOIL stand for and why do I need it?

FOIL stands for First, Outer, Inner, Last - the four multiplications needed when expanding two binomials. It ensures you don't miss any terms and get the complete expanded form.

Q: How do I handle negative signs in binomials?

Treat the negative sign as part of the term. So (x-3) means x + (-3). When you multiply, use sign rules: positive × negative = negative and negative × negative = positive.

Q: Why do some expanded forms have different middle terms?

The middle term comes from the Outer and Inner products in FOIL. For (x+4)(x-3), you get -3x + 4x = +x, but for (x-4)(x-3), you get -3x + (-4x) = -7x.

Q: What if I get the terms in a different order?

That's fine! $ x^2 + x - 12 $ is the same as $ x^2 - 12 + x $. Just make sure you have the correct coefficients for each power of x.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Open parentheses

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

00:16 Calculate the multiplications

00:19 Collect like terms

00:22 This is the simplification for 1, continue to 2

00:25 Open parentheses properly, multiply each term by each term

00:32 Calculate the multiplications

00:39 Collect like terms

00:43 This is the simplification for 2, continue to 3

00:46 Open parentheses properly, multiply each term by each term

00:51 Calculate the multiplications

00:57 Collect like terms

01:00 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Match together the expressions that have the same value

$(x+4)(x-3)$
$(x+4)(x+3)$
$(x-4)(x-3)$
a. $x^2+x-12$
b. $x^2-7x+12$
c. $x^2+7x+12$

2

Step-by-step solution

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

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$ In the formula template for the above distribution law, we take by default that the operation between the terms inside of the parentheses is addition. Note that the sign preceding the term is an inseparable part of it. Furthermore we will apply the laws of sign multiplication to our expression. We will then open the parentheses using the above formula, where there is an addition operation between all terms.

Proceed to simplify each of the expressions in the given problem, whilst making sure to open the parentheses using the mentioned distribution law, the commutative law of addition and multiplication and combining like terms (if there are like terms in the expression obtained after opening the parentheses):

$(x+4)(x-3) \\ \downarrow\\ (x+4)\big(x+(-3)\big) \\ x\cdot x+x\cdot(-3)+4\cdot x+4\cdot(-3)\\ x^2-3x+4x-12\\ \boxed{x^2+x-12}$
$(x+4)(x+3) \\ x\cdot x+x\cdot 3+4\cdot x+4\cdot 3\\ x^2+3x+4x+12\\ \boxed{x^2+7x+12}$
$(x-4)(x-3) \\ \downarrow\\ \big(x+(-4)\big)\big(x+(-3)\big) \\ x\cdot x+x\cdot(-3)+(-4)\cdot x+(-4)\cdot(-3)\\ x^2-3x-4x+12\\ \boxed{x^2-7x+12}$
After applying the commutative law of addition and multiplication we observe that:
The simplified expression in 1 matches the expression in option A,
The simplified expression in 2 matches the expression in option C,
The simplified expression in 3 matches the expression in option B,

Therefore, the correct answer (among the suggested options) is answer B.

3

Final Answer

1-a, 2-c, 3-b

Key Points to Remember

Essential concepts to master this topic

FOIL Rule: First, Outer, Inner, Last terms multiplied systematically
Technique: (x+4)(x-3) = x·x + x·(-3) + 4·x + 4·(-3)
Check: Verify by substituting x=1: both (1+4)(1-3) and 1²+1-12 equal -10 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to apply signs correctly when multiplying
Don't ignore negative signs like treating (x-3) as (x+3) = wrong expanded form! This changes +3x to -3x and gives completely different coefficients. Always treat the sign as part of the term and apply sign rules: positive × negative = negative.

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 why do I need it?

+

FOIL stands for First, Outer, Inner, Last - the four multiplications needed when expanding two binomials. It ensures you don't miss any terms and get the complete expanded form.

How do I handle negative signs in binomials?

+

Treat the negative sign as part of the term. So (x-3) means x + (-3). When you multiply, use sign rules: positive × negative = negative and negative × negative = positive.

Why do some expanded forms have different middle terms?

+

The middle term comes from the Outer and Inner products in FOIL. For (x+4)(x-3), you get -3x + 4x = +x, but for (x-4)(x-3), you get -3x + (-4x) = -7x.

How can I check if my expansion is correct?

+

Pick any value for x (like x=0 or x=1) and substitute it into both the original factored form and your expanded form. If they give the same result, your expansion is correct!

What if I get the terms in a different order?

+

That's fine! $x^2 + x - 12$ is the same as $x^2 - 12 + x$ . Just make sure you have the correct coefficients for each power of x.

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