Sure! Let's go through each part of the problem step-by-step to find the solution.
### Part 1: Simplify the expression [tex]\(4x \cdot 12x^2\)[/tex] and identify any constants
1. [tex]\(4x \cdot 12x^2\)[/tex]:
- Multiply the coefficients: [tex]\(4 \times 12 = 48\)[/tex]
- Combine the powers of [tex]\(x\)[/tex]: [tex]\(x \cdot x^2 = x^{1+2} = x^3\)[/tex]
- Thus, the simplified term is [tex]\(48x^3\)[/tex].
2. The constant provided is [tex]\(8\)[/tex].
### Part 2: Simplify the polynomial [tex]\(6x + 3x^2\)[/tex]
Here, the terms [tex]\(6x\)[/tex] and [tex]\(3x^2\)[/tex] are already simplified and do not need any further simplification.
- [tex]\(6x\)[/tex]
- [tex]\(3x^2\)[/tex]
### Part 3: Simplify the polynomial [tex]\(3.72x^5y^4 + 36x^2y^6 - 54y^3\)[/tex]
In this polynomial, each term is already simplified:
- [tex]\(3.72x^5y^4\)[/tex]
- [tex]\(36x^2y^6\)[/tex]
- [tex]\(-54y^3\)[/tex]
### Summary of the Results:
1. The simplified result of [tex]\(4x \cdot 12x^2\)[/tex] is [tex]\(48x^3\)[/tex] and the constant is [tex]\(8\)[/tex].
2. The polynomial [tex]\(6x + 3x^2\)[/tex] remains [tex]\(6x\)[/tex] and [tex]\(3x^2\)[/tex].
3. The polynomial [tex]\(3.72x^5y^4 + 36x^2y^6 - 54y^3\)[/tex] remains the same as well:
- [tex]\(3.72x^5y^4\)[/tex]
- [tex]\(36x^2y^6\)[/tex]
- [tex]\(-54y^3\)[/tex]
### Conclusion:
To summarize, the relevant results are:
[tex]\[
\begin{aligned}
&\text{1. Simplified expression:} \quad 48x^3 \quad \text{and the constant} \quad 8\\
&\text{2. Polynomial:} \quad 6x \quad \text{and} \quad 3x^2\\
&\text{3. Polynomial:} \quad 3.72x^5y^4, \quad 36x^2y^6, \quad \text{and} \quad -54y^3
\end{aligned}
\][/tex]
Therefore, assembling all these results into the final form:
[tex]\[
(48, 8, 6, 3, 3.72, 36, -54)
\][/tex]
This concludes our detailed step-by-step solution for the given question.
