Louise's answer is incorrect. She misapplied the rules for expanding a binomial raised to a power. Let's go through the expansion step-by-step using the correct method.
### Correct Approach:
The expression [tex]\((5x^3 + 3)^2\)[/tex] is a binomial raised to a power. To expand it correctly, we should use the binomial theorem or recognize the formula for the square of a binomial:
[tex]\[
(a + b)^2 = a^2 + 2ab + b^2,
\][/tex]
where [tex]\(a = 5x^3\)[/tex] and [tex]\(b = 3\)[/tex].
### Step-by-Step Solution:
1. Identify the terms [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- [tex]\(a = 5x^3\)[/tex]
- [tex]\(b = 3\)[/tex]
2. Square each term separately:
- [tex]\(a^2 = (5x^3)^2 = 25x^6\)[/tex]
- [tex]\(b^2 = 3^2 = 9\)[/tex]
3. Compute the middle term, which is [tex]\(2 \cdot a \cdot b\)[/tex]:
- [tex]\(2ab = 2 \cdot (5x^3) \cdot 3 = 2 \cdot 5x^3 \cdot 3 = 30x^3\)[/tex]
4. Combine all the terms:
[tex]\[
(a + b)^2 = a^2 + 2ab + b^2
\][/tex]
Substituting [tex]\(a = 5x^3\)[/tex] and [tex]\(b = 3\)[/tex]:
[tex]\[
(5x^3 + 3)^2 = (5x^3)^2 + 2(5x^3)(3) + 3^2 = 25x^6 + 30x^3 + 9
\][/tex]
### Conclusion:
Therefore, the correct expansion of [tex]\((5x^3 + 3)^2\)[/tex] is:
[tex]\[
\boxed{25x^6 + 30x^3 + 9}
\][/tex]
Louise's answer [tex]\((5x^3 + 3)^2 = 25x^6 + 9\)[/tex] is incorrect because she omitted the middle term [tex]\(2ab = 30x^3\)[/tex].
