To solve the equation $x^2 + 4x = -3$, let's go through the following steps:

Step 1: Move all terms to one side of the equation.

We want the equation in standard quadratic form, $ax^2 + bx + c = 0$. So, add 3 to both sides:

$x^2 + 4x + 3 = 0$

Step 2: Factor the quadratic equation.

Now, we look for two numbers that multiply to give $3$ and add up to $4$. The numbers are $1$ and $3$:

$x^2 + 4x + 3 = (x + 1)(x + 3) = 0$

Step 3: Solve for $x$.

Using the zero-product property, set each factor equal to zero:

$x + 1 = 0 \quad \text{or} \quad x + 3 = 0$

Solving each equation gives:

$x = -1 \quad \text{or} \quad x = -3$

Final Answer:

The solutions are $x = -1$ and $x = -3$.

Would you like more details or have any questions?

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Here are five related questions to consider:

1. What if the constant on the right side of the equation was different, such as $-5$?
2. How would you solve the equation using the quadratic formula instead of factoring?
3. Can all quadratic equations be factored easily, and if not, why?
4. What are some real-world applications of quadratic equations?
5. How does completing the square work to solve quadratic equations?

Tip: Always check your solutions by plugging them back into the original equation to verify that they satisfy the equation!