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To approximate the expression \( (4x^3 + 5)(3x^6 - 8x^2) + 2x^2 + 4x^3 + 13 \) using only the first and last terms, we should identify the first and last terms of each component.
1. The first term of \( 4x^3 + 5 \) is \( 4x^3 \).
2. The first term of \( 3x^6 - 8x^2 \) is \( 3x^6 \).
3. The product of the first terms gives:
\[
(4x^3)(3x^6) = 12x^9
\]
4. The last term of the entire expression \( 2x^2 + 4x^3 + 13 \) is \( 13 \).
So, combining these, the polynomial that best approximates the expression using the first and last terms is:
\[
12x^9 + 13
\]
Thus, the correct polynomial from the given options is:
**12x⁹ + ... + 13**.