To solve this problem, we need to subtract the polynomial $(3x^2 - 5x + 12)$ from $(2x^2 - 3x + 10)$.

Step-by-Step Solution

1. Rewrite the Expression: $(2x^2 - 3x + 10) - (3x^2 - 5x + 12)$

2. Distribute the Negative Sign: $= 2x^2 - 3x + 10 - 3x^2 + 5x - 12$

3. Combine Like Terms:

   • $x^2$ terms: $2x^2 - 3x^2 = -x^2$
   • $x$ terms: $-3x + 5x = 2x$
   • Constant terms: $10 - 12 = -2$

4. Result: $-x^2 + 2x - 2$

Answer

The correct answer is: $\boxed{-x^2 + 2x - 2}$

This corresponds to the fourth option.

Would you like a detailed explanation on a specific step or have any questions? Here are five related questions that might help deepen your understanding:

1. How do you subtract polynomials with different numbers of terms?
2. Why is it necessary to distribute the negative sign when subtracting polynomials?
3. How would this problem change if we were asked to add the polynomials instead?
4. What would happen if we multiplied the two polynomials instead of subtracting?
5. How can you verify the answer by plugging in a value for $x$?

Tip: Always remember to distribute negative signs carefully when subtracting polynomials, as it affects each term in the polynomial being subtracted.