Expand the Expression: (2x+3)(-5-x) Using Distributive Property

Q: Why do I get negative signs everywhere?

Because the second parentheses $ (-5-x) $ contains negative terms! When you multiply positive terms by negative terms, you get negative results. This is normal - just track your signs carefully.

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

Use the FOIL method or make a distribution table. For $ (2x+3)(-5-x) $: multiply every term in the first parentheses by every term in the second.

Q: What does 'like terms' mean exactly?

Like terms have the same variable with the same exponent. So $ -10x $ and $ -3x $ are like terms (both have $ x^1 $), but $ -2x^2 $ is different (it has $ x^2 $).

Q: Why is the final answer arranged from highest to lowest power?

This is standard form for polynomials! We write $ -2x^2 - 13x - 15 $ with the $ x^2 $ term first, then $ x^1 $, then the constant. It makes polynomials easier to read and work with.

Q: How can I check if my expansion is correct?

Pick a simple value like $ x = 1 $. Calculate $ (2(1)+3)(-5-1) = (5)(-6) = -30 $. Then check your answer: $ -2(1)^2 - 13(1) - 15 = -2 - 13 - 15 = -30 $ ✓

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:04 Open parentheses properly, multiply each factor by each factor

00:26 Calculate the multiplications

00:56 Positive times negative always equals negative

01:09 Collect terms

01:17 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

Expand the following expression:

$(2x+3)(-5-x)=$

2

Step-by-step solution

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

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

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.

Therefore the sign preceding the term is an inseparable part of it. We'll also apply the rules of sign multiplication and thus we can present any expression in parentheses, which we'll open using the above formula, first as an expression where addition operation exists between all terms:

$(2x+3)(-5-x)\\ (\textcolor{red}{2x}+\textcolor{blue}{3})((-5)+(-x))\\$ Let's begin by opening the parentheses:

$(\textcolor{red}{2x}+\textcolor{blue}{3})((-5)+(-x))\\ \textcolor{red}{2x}\cdot (-5)+\textcolor{red}{2x}\cdot(-x)+\textcolor{blue}{3}\cdot (-5) +\textcolor{blue}{3} \cdot(-x)\\ -10x-2x^2-15-3x$

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. Like terms are 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 consider its exponent as zero power, due to the fact that raising any number to the power of zero yields the result 1) We'll apply the commutative law of addition, additionally we'll arrange the expression from highest to lowest power from left to right (we'll treat the free number as having zero power):
$\textcolor{purple}{-10x}\textcolor{green}{-2x^2}-15\textcolor{purple}{-3x}\\ \textcolor{green}{-2x^2} \textcolor{purple}{-10x}\textcolor{purple}{-3x}-15\\ \textcolor{green}{-2x^2}\textcolor{purple}{-13x}-15$

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 remains an inseparable part of it,

We therefore got that the correct answer is answer D.

3

Final Answer

$-2x^2-13x-15$

Key Points to Remember

Essential concepts to master this topic

Distribution Rule: Each term multiplies every term in second parentheses
Technique: $2x \cdot (-5) = -10x$ and $2x \cdot (-x) = -2x^2$
Check: Combine like terms carefully: $-10x + (-3x) = -13x$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to distribute both terms to both terms
Don't just multiply $2x \cdot (-5) = -10x$ and stop = incomplete answer! You miss half the terms and get wrong results. Always ensure each term in the first parentheses multiplies every term in the second parentheses.

Practice Quiz

Test your knowledge with interactive questions

$$ (x+y)(x-y)= $$

FAQ

Everything you need to know about this question

Why do I get negative signs everywhere?

+

Because the second parentheses $(-5-x)$ contains negative terms! When you multiply positive terms by negative terms, you get negative results. This is normal - just track your signs carefully.

How do I keep track of all the multiplications?

+

Use the FOIL method or make a distribution table. For $(2x+3)(-5-x)$ : multiply every term in the first parentheses by every term in the second.

What does 'like terms' mean exactly?

+

Like terms have the same variable with the same exponent. So $-10x$ and $-3x$ are like terms (both have $x^1$ ), but $-2x^2$ is different (it has $x^2$ ).

Why is the final answer arranged from highest to lowest power?

+

This is standard form for polynomials! We write $-2x^2 - 13x - 15$ with the $x^2$ term first, then $x^1$ , then the constant. It makes polynomials easier to read and work with.

How can I check if my expansion is correct?

+

Pick a simple value like $x = 1$ . Calculate $(2(1)+3)(-5-1) = (5)(-6) = -30$ . Then check your answer: $-2(1)^2 - 13(1) - 15 = -2 - 13 - 15 = -30$ ✓

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Expand the Expression: (2x+3)(-5-x) Using Distributive Property

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