1. **State the problem:** We are given two numbers, say $x$ and $y$, such that their quotient is $-4$ and their difference is $10$. We need to find the values of $x$ and $y$. 2. **Write the equations:** - Quotient: $$\frac{x}{y} = -4$$ - Difference: $$x - y = 10$$ 3. **Express $x$ in terms of $y$ from the quotient equation:** $$x = -4y$$ 4. **Substitute $x = -4y$ into the difference equation:** $$-4y - y = 10$$ 5. **Simplify the equation:** $$-5y = 10$$ 6. **Solve for $y$:** $$y = \frac{10}{\cancel{-5}} \times \cancel{-1} = -2$$ 7. **Find $x$ using $x = -4y$:** $$x = -4 \times (-2) = 8$$ 8. **Final answer:** The two numbers are $8$ and $-2$.