To solve the quadratic equation $x^2 + 3x = -2$, we first rewrite it in standard form by moving all terms to one side:

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

Now we solve it using factoring.

We need two numbers that multiply to give 2 (the constant term) and add to give 3 (the coefficient of $x$):

• The numbers 1 and 2 work because $1 \times 2 = 2$ and $1 + 2 = 3$.

Thus, we can factor the equation as:

$(x + 1)(x + 2) = 0$

Now, apply the zero-product property by setting each factor equal to zero:

$x + 1 = 0 \quad \text{or} \quad x + 2 = 0$

Solving these:

$x = -1 \quad \text{or} \quad x = -2$

Therefore, the solutions are:

$x = -1 \quad \text{and} \quad x = -2$

Let me know if you would like more details or explanations!

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Here are 5 related questions to explore further:

1. How would you solve a quadratic equation using the quadratic formula?
2. What is the discriminant of a quadratic equation and how is it used?
3. Can all quadratic equations be factored like this?
4. How would the solutions change if the equation were $x^2 + 3x = 2$?
5. What happens when the discriminant is negative?

Tip: When factoring, always check for common factors before attempting more complex methods.