Answer :
We start with the expression
$$
(4x^3 + 9xy + 8y) - (3x^3 + 5xy - 8y).
$$
**Step 1. Distribute the negative sign**
Remember to distribute the minus sign to every term in the second set of parentheses:
$$
4x^3 + 9xy + 8y - 3x^3 - 5xy + 8y.
$$
**Step 2. Combine like terms**
- For the $x^3$ terms:
$$
4x^3 - 3x^3 = x^3.
$$
- For the $xy$ terms:
$$
9xy - 5xy = 4xy.
$$
- For the $y$ terms:
$$
8y + 8y = 16y.
$$
**Step 3. Write the final result**
Combine all the simplified terms:
$$
x^3 + 4xy + 16y.
$$
Thus, the simplified expression is
$$
\boxed{x^3 + 4xy + 16y}.
$$
Comparing with the answer choices, this corresponds to Option C.
$$
(4x^3 + 9xy + 8y) - (3x^3 + 5xy - 8y).
$$
**Step 1. Distribute the negative sign**
Remember to distribute the minus sign to every term in the second set of parentheses:
$$
4x^3 + 9xy + 8y - 3x^3 - 5xy + 8y.
$$
**Step 2. Combine like terms**
- For the $x^3$ terms:
$$
4x^3 - 3x^3 = x^3.
$$
- For the $xy$ terms:
$$
9xy - 5xy = 4xy.
$$
- For the $y$ terms:
$$
8y + 8y = 16y.
$$
**Step 3. Write the final result**
Combine all the simplified terms:
$$
x^3 + 4xy + 16y.
$$
Thus, the simplified expression is
$$
\boxed{x^3 + 4xy + 16y}.
$$
Comparing with the answer choices, this corresponds to Option C.
