The equation provided in the image is:

$x^2 + 4x - 32 = 0$

To solve this quadratic equation, we can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For the equation $x^2 + 4x - 32 = 0$, the coefficients are:

• $a = 1$
• $b = 4$
• $c = -32$

Now, plug in these values into the quadratic formula:

$x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-32)}}{2(1)}$

Simplifying inside the square root:

$x = \frac{-4 \pm \sqrt{16 + 128}}{2}$ $x = \frac{-4 \pm \sqrt{144}}{2}$

Since $\sqrt{144} = 12$, we have:

$x = \frac{-4 \pm 12}{2}$

This gives us two possible solutions:

1. $x_1 = \frac{-4 + 12}{2} = \frac{8}{2} = 4$
2. $x_2 = \frac{-4 - 12}{2} = \frac{-16}{2} = -8$

So the solutions are:

$x = 4 \quad \text{or} \quad x = -8$

Would you like more details on any of the steps or have any other questions?

Here are some related questions for further practice:

1. What is the quadratic formula, and how is it derived?
2. How can you solve a quadratic equation by completing the square?
3. What are the possible types of solutions for a quadratic equation?
4. How can the discriminant ($b^2 - 4ac$) help in understanding the nature of the roots?
5. Can a quadratic equation have complex roots? If so, under what conditions?

Tip: Always check the discriminant ($b^2 - 4ac$) first to determine the number and type of solutions before proceeding with the quadratic formula.