The sum of \(8x - 2\) and \(-3x^2 - 7\) can be found by combining like terms. Here's the step-by-step process:

1. Write the expression:
   \[
   (8x - 2) + (-3x^2 - 7)
   \]

2. Distribute the plus sign (which doesn't change anything in this case):
   \[
   8x - 2 - 3x^2 - 7
   \]

3. Combine like terms:
   - There's only one \(x^2\) term: \(-3x^2\).
   - The \(x\)-term is \(8x\).
   - The constants are \(-2\) and \(-7\), which sum to \(-9\).

Thus, the result is:
\[
-3x^2 + 8x - 9
\]

So, the sum of \(8x - 2\) and \(-3x^2 - 7\) is \(-3x^2 + 8x - 9\).

Would you like further clarification or more details on this? Here are five related questions:

1. How do you add polynomials with more than one variable?
2. Can you subtract these two expressions and see the result?
3. What happens if we multiply \(8x - 2\) and \(-3x^2 - 7\)?
4. How do you factor quadratic expressions like \(-3x^2 + 8x - 9\)?
5. Can you find the roots of \(-3x^2 + 8x - 9 = 0\) using the quadratic formula?

**Tip:** Always align like terms when adding or subtracting polynomials to simplify the process.

Learn how to add the polynomials 8x - 2 and -3x^2 - 7 with a clear step-by-step explanation. This algebraic problem demonstrates combining like terms and applying the commutative property of addition. Perfect for students in Grades 8-10.

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