1. **State the problem:** We need to divide the polynomial $-3x^2 + 68$ by the binomial $x - 5$. 2. **Formula and method:** Polynomial division can be done using long division or synthetic division. Here, we use long division. 3. **Set up the division:** Divide the leading term of the numerator $-3x^2$ by the leading term of the denominator $x$ to get the first term of the quotient: $$\frac{-3x^2}{x} = -3x.$$ 4. **Multiply and subtract:** Multiply $-3x$ by $x - 5$ to get $-3x^2 + 15x$. Subtract this from the original polynomial: $$(-3x^2 + 68) - (-3x^2 + 15x) = 0 - 15x + 68 = -15x + 68.$$ 5. **Repeat the process:** Divide the new leading term $-15x$ by $x$ to get $-15$. Multiply $-15$ by $x - 5$ to get $-15x + 75$. Subtract this from the remainder: $$(-15x + 68) - (-15x + 75) = 0 - 7 = -7.$$ 6. **Conclusion:** The quotient is $-3x - 15$ and the remainder is $-7$. So the division can be expressed as: $$\frac{-3x^2 + 68}{x - 5} = -3x - 15 + \frac{-7}{x - 5}.$$