Which of the following polynomials can approximate the expression (4x3+5)(3x6−8x2)2x2+4x3+13 using first and last terms only?(1 point) Responses 12x^9+...13 6x7+...4x^^3 4x^3+...13 6x^7+...−7

To approximate the expression \((4x^3 + 5)(3x^6 - 8x^2)2x^2 + 4x^3 + 13\) using only the first and last terms, we need to consider the leading (highest degree) and the constant (last term) from the expression.

1. **Identify the first term**: We focus on the leading terms from both polynomials.

- The first polynomial is \(4x^3 + 5\). The leading term is \(4x^3\).
- The second polynomial is \(3x^6 - 8x^2\). The leading term is \(3x^6\).

Therefore, the leading term from the product \((4x^3)(3x^6)\) is:
\[
4x^3 \cdot 3x^6 = 12x^9.
\]

2. **Identify the last term**: The constant term in the expression is \(+13\). The term comes from the constant \(5\) in the first polynomial, as well as the constant term of the second polynomial, which is \(-8x^2\) times \(2x^2\) does not contribute a constant to the final sum. The constant \(5\) remains and contributes nothing to the overall leading power apart from the addition.

Now, we can approximate the expression with just the leading and constant terms:
\[
12x^9 + 13.
\]

Given this analysis, among the provided options, the correct polynomial that represents this approximation would be:
\[
12x^9 + ... + 13.
\]

Thus, the answer is:
**12x^9 + ... + 13.**