To find a quadratic equation where the sum and product of the roots are given, we can use the general form of a quadratic equation based on its roots:

$x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0$

In this case:

• The sum of the roots is $-3$.
• The product of the roots is $2$.

Substituting these values into the equation:

$x^2 - (-3)x + 2 = 0$

This simplifies to:

$x^2 + 3x + 2 = 0$

Thus, the quadratic equation with a sum of roots $-3$ and a product of roots $2$ is:

$x^2 + 3x + 2 = 0$

Would you like a breakdown of how to factor this equation or further details?

Here are five related questions for further exploration:

1. What are the actual roots of the equation $x^2 + 3x + 2 = 0$?
2. How can we verify the sum and product of the roots of $x^2 + 3x + 2 = 0$?
3. How would the equation change if the sum and product of the roots were different values?
4. Can you find a quadratic equation with roots that have a sum of 4 and a product of $-5$?
5. How does the quadratic formula confirm the roots for $x^2 + 3x + 2 = 0$?

Tip: Remember that if you know the sum and product of the roots, you can always construct a quadratic equation in the form $x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0$.