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

$(3+20)\times(12+4)=$

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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