The image shows the following mathematical expression:

$\sqrt{x^2 + 2x + 8} = 12 - 2x$

We need to solve for $x$.

Steps to solve:

1. Square both sides to eliminate the square root: $(\sqrt{x^2 + 2x + 8})^2 = (12 - 2x)^2$ This simplifies to: $x^2 + 2x + 8 = (12 - 2x)^2$

2. Expand the right-hand side: $(12 - 2x)^2 = 144 - 48x + 4x^2$ So, the equation becomes: $x^2 + 2x + 8 = 144 - 48x + 4x^2$

3. Move all terms to one side: Bring all terms to the left-hand side: $x^2 + 2x + 8 - 144 + 48x - 4x^2 = 0$ Simplify the terms: $-3x^2 + 50x - 136 = 0$

4. Solve the quadratic equation using the quadratic formula: The quadratic equation is in the form $ax^2 + bx + c = 0$, where: $a = -3, \, b = 50, \, c = -136$ The quadratic formula is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Substituting the values of $a$, $b$, and $c$: $x = \frac{-50 \pm \sqrt{50^2 - 4(-3)(-136)}}{2(-3)}$ Simplifying inside the square root: $x = \frac{-50 \pm \sqrt{2500 - 1632}}{-6}$ $x = \frac{-50 \pm \sqrt{868}}{-6}$ $x = \frac{-50 \pm 29.45}{-6}$

5. Find the two possible solutions: $x_1 = \frac{-50 + 29.45}{-6} \approx \frac{-20.55}{-6} \approx 3.425$ $x_2 = \frac{-50 - 29.45}{-6} \approx \frac{-79.45}{-6} \approx 13.24$

Thus, the two solutions for $x$ are approximately: $x \approx 3.425 \quad \text{or} \quad x \approx 13.24$

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

Here are 5 related questions you might consider:

1. How do we check if both solutions satisfy the original equation?
2. What happens if we change the constants in the equation?
3. Can we solve this quadratic equation graphically? What would the graph look like?
4. What is the significance of the discriminant in the quadratic formula?
5. Can this method be applied to similar equations involving square roots?

Tip: Always check both solutions by substituting them back into the original equation to ensure they are valid!