Answer :
Louise's attempt to expand the expression [tex]\((5x^3 + 3)^2\)[/tex] was incorrect. Let's go through the correct process step-by-step.
The expression [tex]\((5x^3 + 3)^2\)[/tex] can be expanded using the formula for the square of a binomial:
[tex]\[
(a + b)^2 = a^2 + 2ab + b^2
\][/tex]
In this formula, let [tex]\(a = 5x^3\)[/tex] and [tex]\(b = 3\)[/tex]. Now, we'll apply the formula:
1. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
a^2 = (5x^3)^2 = 25x^6
\][/tex]
2. Calculate [tex]\(2ab\)[/tex]:
[tex]\[
2ab = 2 \times (5x^3) \times 3 = 2 \times 5 \times 3 \times x^3 = 30x^3
\][/tex]
3. Calculate [tex]\(b^2\)[/tex]:
[tex]\[
b^2 = 3^2 = 9
\][/tex]
4. Combine all the terms:
[tex]\[
(5x^3 + 3)^2 = a^2 + 2ab + b^2 = 25x^6 + 30x^3 + 9
\][/tex]
Therefore, the correct expanded form of [tex]\((5x^3 + 3)^2\)[/tex] is:
[tex]\[ 25x^6 + 30x^3 + 9 \][/tex]
The expression [tex]\((5x^3 + 3)^2\)[/tex] can be expanded using the formula for the square of a binomial:
[tex]\[
(a + b)^2 = a^2 + 2ab + b^2
\][/tex]
In this formula, let [tex]\(a = 5x^3\)[/tex] and [tex]\(b = 3\)[/tex]. Now, we'll apply the formula:
1. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
a^2 = (5x^3)^2 = 25x^6
\][/tex]
2. Calculate [tex]\(2ab\)[/tex]:
[tex]\[
2ab = 2 \times (5x^3) \times 3 = 2 \times 5 \times 3 \times x^3 = 30x^3
\][/tex]
3. Calculate [tex]\(b^2\)[/tex]:
[tex]\[
b^2 = 3^2 = 9
\][/tex]
4. Combine all the terms:
[tex]\[
(5x^3 + 3)^2 = a^2 + 2ab + b^2 = 25x^6 + 30x^3 + 9
\][/tex]
Therefore, the correct expanded form of [tex]\((5x^3 + 3)^2\)[/tex] is:
[tex]\[ 25x^6 + 30x^3 + 9 \][/tex]
