To simplify the expression $(4x - 6x^2 - 7) - (2 + 7x - 5x^2)$, let’s go through the steps carefully.

Step 1: Distribute the negative sign

Rewrite the expression by distributing the negative sign across the terms in the second parenthesis: $(4x - 6x^2 - 7) - 2 - 7x + 5x^2$

Step 2: Combine like terms

Now, group and simplify terms with the same power of $x$:

• $x^2$ terms: $-6x^2 + 5x^2 = -x^2$
• $x$ terms: $4x - 7x = -3x$
• Constant terms: $-7 - 2 = -9$

Simplified Expression

Putting it all together, we get: $-x^2 - 3x - 9$

Conclusion

The correct answer is: D) $-x^2 - 3x - 9$

Would you like further details or have any other questions?

Here are some related questions you might find helpful:

1. What are the general steps for simplifying algebraic expressions with multiple terms?
2. How do you determine the correct sign when distributing in expressions?
3. What are some common mistakes to avoid when combining like terms?
4. How can you verify your solution to a simplification problem?
5. Can this method be applied to polynomials of higher degrees?

Tip: Always check each step when distributing and combining terms to avoid errors with signs.