Which of the following polynomials can approximate the expression (4x3+5)(3x6−8x2)2x2 +4x3+13

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

1. **Leading term**:
- The leading term of \( 4x^3 \) is \( 4x^3 \).
- The leading term of \( 5 \) is simply \( 5 \).
- The leading term of \( 3x^6 \) is \( 3x^6 \).
- The leading term of \( -8x^2 \) is \( -8x^2 \).
- The leading term of \( 2x^2 \) is \( 2x^2 \).

When we multiply these together to get the leading term of the entire expression, we focus on the largest degree:
\[
(4x^3)(3x^6)(2x^2) = 24x^{11}.
\]

2. **Constant term**:
- From \( 4x^3 + 5 \), the constant term is \( 5 \).
- From \( 3x^6 - 8x^2 \), the constant term is \( 0 \).
- The constant term from \( 2x^2 \) is also \( 0 \).
- The additional terms give us \( 4x^3 \) which has no constant term and \( 13 \), a constant term.

So the only constant term coming into play from the entire expression is \( 13 \).

Putting this together, the leading term is \( 24x^{11} \) and the constant term is \( 13 \).

Given the options in the responses:

1. \( 6x^7 + ... + 4x^3 \) (not right).
2. \( 12x^9 + ... + 13 \) (not right).
3. \( 6x^7 + ... - 7 \) (not right).
4. \( 4x^3 + ... + 13 \) (not right).

None of the options correctly reflect the leading term of \( 24x^{11} \) and the constant term as \( 13 \).

The answer cannot be determined accurately with the given options since they do not include \( 24x^{11} \). Thus, if you have to choose one that includes a constant term with \( 13 \), you would select the fourth option \( 4x^3 + ... + 13 \) as it includes the relevant constant term despite its leading term not being correct.