Question: Use long division to divide the polynomial $-30x^6 + 48x^5 + 50x - 80$ by $5x - 8$. Write your answer in standard form.
1. **State the problem:** Divide the polynomial $$-30x^6 + 48x^5 + 50x - 80$$ by the binomial $$5x - 8$$ using long division.
2. **Recall the formula and rules:** Polynomial long division is similar to numerical long division. We divide the leading term of the dividend by the leading term of the divisor, multiply the divisor by this quotient term, subtract from the dividend, and repeat with the remainder.
3. **Arrange terms in descending powers and include missing terms:**
Dividend: $$-30x^6 + 48x^5 + 0x^4 + 0x^3 + 0x^2 + 50x - 80$$
Divisor: $$5x - 8$$
4. **Divide leading terms:**
$$\frac{-30x^6}{5x} = -6x^5$$
5. **Multiply divisor by $$-6x^5$$:**
$$-6x^5 \times (5x - 8) = -30x^6 + 48x^5$$
6. **Subtract:**
$$(-30x^6 + 48x^5) - (-30x^6 + 48x^5) = 0$$
Bring down the next terms: $$0x^4 + 0x^3 + 0x^2 + 50x - 80$$
7. **Divide leading term of new dividend by leading term of divisor:**
$$\frac{0x^4}{5x} = 0x^3$$
Multiply divisor by 0: $$0$$
Subtract: no change.
Bring down next terms.
8. **Repeat for $$0x^3$$ and $$0x^2$$ terms:**
All zero, so quotient terms are zero.
9. **Divide $$50x$$ by $$5x$$:**
$$\frac{50x}{5x} = 10$$
10. **Multiply divisor by 10:**
$$10 \times (5x - 8) = 50x - 80$$
11. **Subtract:**
$$(50x - 80) - (50x - 80) = 0$$
12. **No remainder left.**
13. **Write the quotient:**
$$-6x^5 + 0x^4 + 0x^3 + 0x^2 + 10 = -6x^5 + 10$$
**Final answer:** $$\boxed{-6x^5 + 10}$$