Verify the Expansion: (4x+3)(8x+5) = 32x²+44x+15

Q: What does FOIL actually stand for?

First terms, Outer terms, Inner terms, Last terms. For $ (4x+3)(8x+5) $: First = $ 4x \cdot 8x $, Outer = $ 4x \cdot 5 $, Inner = $ 3 \cdot 8x $, Last = $ 3 \cdot 5 $.

Q: Why do I need to combine like terms at the end?

After FOIL, you get four separate terms that might include like terms (same variable and power). You must combine $ 20x + 24x = 44x $ to get the final simplified form.

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

Substitute a simple value like x = 1 into both sides. For our problem: $ (4+3)(8+5) = 7 \times 13 = 91 $ and $ 32 + 44 + 15 = 91 $ ✓

Q: What if I get confused with the signs?

Be extra careful with positive and negative signs! Write each step clearly and remember that multiplying two negatives gives a positive result.

Q: Is there a pattern to remember for squaring binomials?

Yes! For $ (a+b)^2 $, the pattern is $ a^2 + 2ab + b^2 $. But this problem involves different binomials, so stick with FOIL for safety.

Polynomial Multiplication with Binomial Expansion

Is equality correct?

$(4x+3)(8x+5)=32x^2+44x+15$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Are the expressions equal?

00:03 Let's properly open parentheses, multiply each factor by each factor

00:24 Let's calculate the multiplications

00:40 Let's group the factors

00:48 Let's compare the expression terms, we can see they're equal

00:52 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?

$(4x+3)(8x+5)=32x^2+44x+15$

Step-by-step solution

To solve this problem, we'll employ the distributive property to verify the given equality:

Step 1: Expand the left-hand side expression $(4x+3)(8x+5)$ :

Calculate $4x \cdot 8x = 32x^2$
Calculate $4x \cdot 5 = 20x$
Calculate $3 \cdot 8x = 24x$
Calculate $3 \cdot 5 = 15$

Step 2: Combine like terms:

The expanded form is:
$32x^2 + 20x + 24x + 15$

Combine like terms $20x$ and $24x$ :

This gives us $32x^2 + (20x + 24x) + 15 = 32x^2 + 44x + 15$ .

Step 3: Compare the expanded form with the right-hand side expression:

The expanded form, $32x^2 + 44x + 15$ , matches the right-hand side exactly.

Thus, the given equality $(4x+3)(8x+5) = 32x^2 + 44x + 15$ is correct.

The correct answer 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: $4x \cdot 8x = 32x^2$ , then $4x \cdot 5 + 3 \cdot 8x = 44x$
Verification: Check all four products sum correctly: $32x^2 + 20x + 24x + 15 = 32x^2 + 44x + 15$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to multiply all four term combinations
Don't just multiply first terms and last terms = $32x^2 + 15$ ! This misses the middle terms completely. Always use FOIL: multiply First + Outer + Inner + Last to get all four products.

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

What does FOIL actually stand for?

First terms, Outer terms, Inner terms, Last terms. For $(4x+3)(8x+5)$ : First = $4x \cdot 8x$ , Outer = $4x \cdot 5$ , Inner = $3 \cdot 8x$ , Last = $3 \cdot 5$ .

Why do I need to combine like terms at the end?

After FOIL, you get four separate terms that might include like terms (same variable and power). You must combine $20x + 24x = 44x$ to get the final simplified form.

How can I check if my expansion is correct?

Substitute a simple value like x = 1 into both sides. For our problem: $(4+3)(8+5) = 7 \times 13 = 91$ and $32 + 44 + 15 = 91$ ✓

What if I get confused with the signs?

Be extra careful with positive and negative signs! Write each step clearly and remember that multiplying two negatives gives a positive result.

Is there a pattern to remember for squaring binomials?

Yes! For $(a+b)^2$ , the pattern is $a^2 + 2ab + b^2$ . But this problem involves different binomials, so stick with FOIL for safety.

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