Expand (2x+4)(x-5): Find the Missing Term in 2x²-6x+☐

Q: Why do I need to multiply every term by every other term?

The distributive property requires you to multiply each term in the first parentheses by each term in the second. Missing any multiplication gives an incomplete expansion!

Q: How do I handle the negative signs correctly?

Remember that the sign is part of the term! When you see (x-5), treat it as (x + (-5)). Then multiply signs: positive × negative = negative.

Q: What if I get confused with all the multiplications?

Use the FOIL method for binomials: First terms: 2x × xOuter terms: 2x × (-5)Inner terms: 4 × xLast terms: 4 × (-5)

Q: How do I combine like terms at the end?

Look for terms with the same variable and exponent. In this problem: -10x + 4x = -6x. The $ x^2 $ and constant terms stay separate.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Let's fill in the missing parts together.

00:12 We'll use shortened multiplication formulas to help us out.

00:19 First, we'll match the numbers to the correct variables. Like a puzzle!

00:25 Next, we'll substitute them in to find what's missing.

00:29 And that's how we solve this question. Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Complete the following equation:

_{$(2x+4)(x-5)=2x^2-6x+\textcolor{red}{☐}$}

2

Step-by-step solution

Simplify the given expression on the left side:

$(2x+4)(x-5)$ Open the parentheses by applying the expanded 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 distribution law mentioned, the operation between terms inside of the parentheses is automatically addition. Remember that the sign preceding a term is an inseparable part of it. We'll apply the rules of sign multiplication allowing us to present any expression inside of the parentheses. The parentheses will be opened by using the above formula, first, as an expression where addition occurs between all terms (if needed),

Therefore, we'll first present each of the expressions inside of the parentheses in the multiplication on the left side as an expression containing addition:

$(2x+4)(x-5)=2x^2-6x+\textcolor{purple}{\boxed{?}} \\ \downarrow\\ \big(2x+(+4)\big)\big(x+(-5)\big)=2x^2-6x+\textcolor{purple}{\boxed{?}} \\$ Let's once again write the expanded distribution law mentioned earlier:

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

And apply it to our problem:

$\big(\textcolor{red}{2x}+\textcolor{blue}{(+4)}\big)\big(x+(-5)\big)=2x^2-6x+\textcolor{purple}{\boxed{?}} \\ \downarrow\\ \textcolor{red}{2x}\cdot x +\textcolor{red}{2x}\cdot (-5)+\textcolor{blue}{(+4)}\cdot x +\textcolor{blue}{(+4)}\cdot (-5)=2x^2-6x+\textcolor{purple}{\boxed{?}} \\$ Continue to apply the multiplication sign rules. Remember that multiplying terms with identical signs yields a positive result, and multiplying terms with different signs yields a negative result. In the next step we'll combine like terms in the expression obtained on the left side:

$\textcolor{red}{2x}\cdot x +\textcolor{red}{2x}\cdot (-5)+\textcolor{blue}{(+4)}\cdot x +\textcolor{blue}{(+4)}\cdot (-5)=2x^2-6x+\textcolor{purple}{\boxed{?}} \\ \downarrow\\ 2x^2-10x+4x-20=2x^2-6x+\textcolor{purple}{\boxed{?}} \\ 2x^2-6x-20=2x^2-6x+\textcolor{purple}{\boxed{?}}$

Therefore, according to the multiplication sign rules, the missing expression is the number $-20$ ,

In other words - the correct answer is C.

3

Final Answer

20-

Key Points to Remember

Essential concepts to master this topic

Distribution Rule: Apply (a+b)(c+d) = ac + ad + bc + bd
Technique: Calculate 2x·(-5) = -10x and 4·(-5) = -20
Check: Combine like terms to verify 2x² - 10x + 4x - 20 = 2x² - 6x - 20 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to include all four terms from distribution
Don't multiply just (2x)(x) and (4)(-5) = 2x² - 20! This skips the middle terms and gives the wrong answer. Always multiply every term in the first parentheses by every term in the second: 2x·x + 2x·(-5) + 4·x + 4·(-5).

Practice Quiz

Test your knowledge with interactive questions

$$ x^2+6x+9=0 $$

What is the value of X?

FAQ

Everything you need to know about this question

Why do I need to multiply every term by every other term?

+

The distributive property requires you to multiply each term in the first parentheses by each term in the second. Missing any multiplication gives an incomplete expansion!

How do I handle the negative signs correctly?

+

Remember that the sign is part of the term! When you see (x-5), treat it as (x + (-5)). Then multiply signs: positive × negative = negative.

What if I get confused with all the multiplications?

+

Use the FOIL method for binomials:

First terms: 2x × x
Outer terms: 2x × (-5)
Inner terms: 4 × x
Last terms: 4 × (-5)

How do I combine like terms at the end?

+

Look for terms with the same variable and exponent. In this problem: -10x + 4x = -6x. The $x^2$ and constant terms stay separate.

Why is my final answer negative?

+

The constant term 4 × (-5) = -20 because you're multiplying a positive by a negative. Always check your sign rules when multiplying!

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Expand (2x+4)(x-5): Find the Missing Term in 2x²-6x+☐

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