Verify the Equation: (-4-x)(7+x) = -28-11x-x²

Q: Why do we use FOIL instead of just distributing?

FOIL is organized distribution! It ensures you multiply every term in the first binomial by every term in the second. This prevents missing terms like $ (-x)(x) $.

Q: How do I handle the negative signs correctly?

Think of $ (-4-x) $ as $ (-4) + (-x) $. When multiplying negatives: negative × positive = negative and negative × negative = positive.

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

Order doesn't matter for addition! $ -28 - 11x - x^2 $ is the same as $ -x^2 - 11x - 28 $. Just make sure all coefficients and signs match.

Q: Can I check my answer by substituting a number for x?

Yes! Try $ x = 1 $: Left side: $ (-4-1)(7+1) = (-5)(8) = -40 $. Right side: $ -28-11(1)-1^2 = -40 $. They match!

Q: Why is the x² term negative in the final answer?

Because $ (-x) \times (x) = -x^2 $. The negative sign from $ -x $ makes the squared term negative, not positive!

Polynomial Expansion with Distributive Property

Is equality correct?

$(-4-x)(7+x)=-28-11x-x^2$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:10 Are these expressions equal? Let's find out!

00:14 First, open the parentheses. Multiply each factor by every other factor. Ready?

00:32 Now, calculate each multiplication carefully. You can do it!

00:49 Great! Let's group all the similar factors together. Excellent!

00:56 Finally, compare the terms of both expressions. Are they the same? Yes, they are!

01:02 And that's how we find the solution. Nice work!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Is equality correct?

$(-4-x)(7+x)=-28-11x-x^2$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Expand the expression on the left side using the distributive property
Step 2: Simplify the expanded expression
Step 3: Compare the simplified expression with the given expression on the right side

Now, let's work through each step:
Step 1: Expand using the distributive property. We apply the formula: $(a+b)(c+d) = ac + ad + bc + bd$ .

For $(-4-x)(7+x)$ , distribute each term:
$(-4)(7) + (-4)(x) + (-x)(7) + (-x)(x)$

Step 2: Simplify the terms:
$-28 - 4x - 7x - x^2$
Combine like terms: $-28 - 11x - x^2$

Step 3: Compare this with the given expression $-28 - 11x - x^2$ . Both sides of the equation are identical, indicating the expressions are equivalent.

Therefore, the solution to the problem confirms the expressions are equal, and the correct choice is Yes.

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic

FOIL Method: First, Outer, Inner, Last terms multiply systematically
Technique: $(-4)(7) + (-4)(x) + (-x)(7) + (-x)(x) = -28 - 4x - 7x - x^2$
Check: Combine like terms and compare both sides exactly ✓

Common Mistakes

Avoid these frequent errors

Forgetting negative signs during multiplication
Don't ignore negative signs when multiplying = wrong coefficients! $(-x)(x) = -x^2$ not $+x^2$ . This changes the entire result. Always track each negative sign carefully through every multiplication step.

Practice Quiz

Test your knowledge with interactive questions

It is possible to use the distributive property to simplify the expression below?

What is its simplified form?

$$ (ab)(c d) $$

$$

FAQ

Everything you need to know about this question

Why do we use FOIL instead of just distributing?

FOIL is organized distribution! It ensures you multiply every term in the first binomial by every term in the second. This prevents missing terms like $(-x)(x)$ .

How do I handle the negative signs correctly?

Think of $(-4-x)$ as $(-4) + (-x)$ . When multiplying negatives: negative × positive = negative and negative × negative = positive.

What if I get the terms in a different order?

Order doesn't matter for addition! $-28 - 11x - x^2$ is the same as $-x^2 - 11x - 28$ . Just make sure all coefficients and signs match.

Can I check my answer by substituting a number for x?

Yes! Try $x = 1$ : Left side: $(-4-1)(7+1) = (-5)(8) = -40$ . Right side: $-28-11(1)-1^2 = -40$ . They match!

Why is the x² term negative in the final answer?

Because $(-x) \times (x) = -x^2$ . The negative sign from $-x$ makes the squared term negative, not positive!

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