The problem states:

"Two numbers multiply together to give $-20$ and add together to give $-1$. What are the two numbers?"

Let's solve this step by step.

We need to find two numbers, say $x$ and $y$, that satisfy:

1. $x \times y = -20$
2. $x + y = -1$

To find these numbers, we can use the following approach:

Step 1: Express one variable in terms of the other

From the second equation, we can express $y$ in terms of $x$: $y = -1 - x$

Step 2: Substitute into the first equation

Now, substitute this expression for $y$ into the first equation: $x \times (-1 - x) = -20$ Simplify: $-x - x^2 = -20$ Rearrange this to get a quadratic equation: $x^2 + x - 20 = 0$

Step 3: Solve the quadratic equation

Now, we solve the quadratic equation $x^2 + x - 20 = 0$. We can use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ For this equation, $a = 1$, $b = 1$, and $c = -20$.

Substitute these values into the quadratic formula: $x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-20)}}{2(1)}$ $x = \frac{-1 \pm \sqrt{1 + 80}}{2}$ $x = \frac{-1 \pm \sqrt{81}}{2}$ $x = \frac{-1 \pm 9}{2}$

Step 4: Calculate the two solutions

The two possible values for $x$ are:

1. $x = \frac{-1 + 9}{2} = \frac{8}{2} = 4$
2. $x = \frac{-1 - 9}{2} = \frac{-10}{2} = -5$

Step 5: Find the corresponding values for $y$

For $x = 4$, $y = -1 - 4 = -5$. For $x = -5$, $y = -1 - (-5) = 4$.

Thus, the two numbers are $4$ and $-5$.

Would you like more details or have any other questions?

Here are 5 related questions to expand on this topic:

1. How would you solve a quadratic equation by factoring?
2. Can you think of another pair of numbers that multiply to a different negative product but add to $-1$?
3. How do you use the discriminant to determine the number of real solutions for a quadratic equation?
4. What are the steps to complete the square for solving quadratic equations?
5. How can you apply the quadratic formula to solve more complex equations?

Tip: Always check your solutions by substituting them back into the original equations!