Verify the Equation: (-2a+3b)(4c+5a) = 8ac+10a²-12bc-15ab

Q: Why are all the signs opposite in my answer?

This means you calculated correctly! When expanding $ (-2a+3b)(4c+5a) $, you get $ -8ac-10a^2+12bc+15ab $, which is exactly the negative of the given expression.

Q: How do I remember which terms to multiply together?

Use FOIL: First terms, Outer terms, Inner terms, Last terms. For $ (-2a+3b)(4c+5a) $:First: $ (-2a)(4c) = -8ac $Outer: $ (-2a)(5a) = -10a^2 $Inner: $ (3b)(4c) = 12bc $Last: $ (3b)(5a) = 15ab $

Q: What does it mean when the expression equals the negative?

It means the given equation is incorrect. The left side equals $ -(\text{right side}) $. This is a common type of verification problem where you check if an algebraic identity is true or false.

Q: Should I always expand the left side completely?

Yes! Always expand fully and combine like terms before comparing. Don't try to match terms before completing all the multiplication - you might miss sign errors or incorrectly combine terms.

Q: How can I avoid sign errors when multiplying?

Write out each multiplication step clearly:Positive × Positive = PositiveNegative × Positive = NegativePositive × Negative = NegativeNegative × Negative = PositiveTake your time with each step!

Polynomial Multiplication with Sign Verification

Is equality correct?

$(-2a+3b)(4c+5a)\stackrel{?}{=}8ac+10a^2-12bc-15ab$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Are the expressions equal?

00:04 Let's properly open parentheses, multiply each factor by each factor

00:24 Let's calculate the multiplications

00:47 We'll take out the minus and see that then the expressions are equal, therefore they are not equal

00:54 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Is equality correct?

$(-2a+3b)(4c+5a)\stackrel{?}{=}8ac+10a^2-12bc-15ab$

Step-by-step solution

To determine if the given algebraic expression is correct, we will expand the left-hand side using the distributive property:

The expression is $(-2a + 3b)(4c + 5a)$ .

Step-by-step expansion:

Multiply $-2a$ by $4c$ : $-2a \times 4c = -8ac$ .
Multiply $-2a$ by $5a$ : $-2a \times 5a = -10a^2$ .
Multiply $3b$ by $4c$ : $3b \times 4c = 12bc$ .
Multiply $3b$ by $5a$ : $3b \times 5a = 15ab$ .

Combine these results: $-8ac - 10a^2 + 12bc + 15ab$ .

Now compare this result with the right-hand side of the given expression $8ac + 10a^2 - 12bc - 15ab$ .

We can observe that each corresponding term has the opposite sign.

This shows that the original statement is incorrect.

Therefore, the expression is actually the negative of what was given, so:

The expression is exactly the same as $-(8ac + 10a^2 - 12bc - 15ab)$ .

Thus, the correct choice is:

No, the expression is exactly the same as $-(8ac + 10a^2 - 12bc - 15ab)$ .

Final Answer

No, the expression is exactly the same as $-(8ac+10a^2-12bc-15ab)$

Key Points to Remember

Essential concepts to master this topic

FOIL Method: Distribute each term to get all four products
Technique: $(-2a)(4c) = -8ac$ and $(3b)(5a) = 15ab$
Check: Compare each term's sign in your result with given expression ✓

Common Mistakes

Avoid these frequent errors

Forgetting to distribute negative signs correctly
Don't forget that $(-2a)(4c) = -8ac$ , not $8ac$ ! Missing negative signs gives completely opposite results. Always track each negative sign 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 are all the signs opposite in my answer?

This means you calculated correctly! When expanding $(-2a+3b)(4c+5a)$ , you get $-8ac-10a^2+12bc+15ab$ , which is exactly the negative of the given expression.

How do I remember which terms to multiply together?

Use FOIL: First terms, Outer terms, Inner terms, Last terms. For $(-2a+3b)(4c+5a)$ :

First: $(-2a)(4c) = -8ac$
Outer: $(-2a)(5a) = -10a^2$
Inner: $(3b)(4c) = 12bc$
Last: $(3b)(5a) = 15ab$

What does it mean when the expression equals the negative?

It means the given equation is incorrect. The left side equals $-(\text{right side})$ . This is a common type of verification problem where you check if an algebraic identity is true or false.

Should I always expand the left side completely?

Yes! Always expand fully and combine like terms before comparing. Don't try to match terms before completing all the multiplication - you might miss sign errors or incorrectly combine terms.

How can I avoid sign errors when multiplying?

Write out each multiplication step clearly:

Positive × Positive = Positive
Negative × Positive = Negative
Positive × Negative = Negative
Negative × Negative = Positive

Take your time with each step!

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