Question: Simplify each expression by Adding or Subtracting. (Chapter 7)
$$(2x^2 + 7x^3 - 5x^4) - (4x^4 + 2x^2 - 5x^3)$$
1. **State the problem:** Simplify the expression $$(2x^2 + 7x^3 - 5x^4) - (4x^4 + 2x^2 - 5x^3)$$ by adding or subtracting like terms.
2. **Write the expression clearly:**
$$2x^2 + 7x^3 - 5x^4 - 4x^4 - 2x^2 + 5x^3$$
3. **Group like terms:**
$$(2x^2 - 2x^2) + (7x^3 + 5x^3) + (-5x^4 - 4x^4)$$
4. **Simplify each group:**
- For $x^2$ terms: $$2x^2 - 2x^2 = \cancel{2x^2} - \cancel{2x^2} = 0$$
- For $x^3$ terms: $$7x^3 + 5x^3 = 12x^3$$
- For $x^4$ terms: $$-5x^4 - 4x^4 = -9x^4$$
5. **Combine the simplified terms:**
$$0 + 12x^3 - 9x^4 = -9x^4 + 12x^3$$
6. **Final answer:**
$$\boxed{-9x^4 + 12x^3}$$
This matches the third option in the multiple-choice list.