Let's solve the inequality $x^2 + 6x \geq -8$.

Step 1: Rearrange the inequality

First, move $-8$ to the other side to set the inequality to $\geq 0$:

$x^2 + 6x + 8 \geq 0$

Step 2: Factor the quadratic expression

We now factor $x^2 + 6x + 8$. The factors of 8 that sum to 6 are 4 and 2, so:

$x^2 + 6x + 8 = (x + 2)(x + 4)$

Thus, the inequality becomes:

$(x + 2)(x + 4) \geq 0$

Step 3: Find the critical points

Set the factors equal to zero to find the critical points:

$x + 2 = 0 \quad \text{gives} \quad x = -2$ $x + 4 = 0 \quad \text{gives} \quad x = -4$

These are the critical points where the expression equals zero.

Step 4: Analyze the intervals

The critical points split the number line into three intervals:

1. $(-\infty, -4)$
2. $(-4, -2)$
3. $(-2, \infty)$

We will test points from each interval to determine where the expression is positive or negative.

• For $x = -5$ (in $(-\infty, -4)$): $(-5 + 2)(-5 + 4) = (-3)(-1) = 3 \quad (\text{positive})$

• For $x = -3$ (in $(-4, -2)$): $(-3 + 2)(-3 + 4) = (-1)(1) = -1 \quad (\text{negative})$

• For $x = 0$ (in $(-2, \infty)$): $(0 + 2)(0 + 4) = 2 \times 4 = 8 \quad (\text{positive})$

Step 5: Determine the solution

The expression is non-negative (either positive or zero) in the intervals:

• $(-\infty, -4]$
• $[-2, \infty)$

Thus, the solution to the inequality is:

$x \in (-\infty, -4] \cup [-2, \infty)$

Step 6: Graph the solution

On the number line:

• There will be a filled circle at $x = -4$ and at $x = -2$ since both points are included in the solution (due to the $\geq$ sign).
• The intervals $(-\infty, -4]$ and $[-2, \infty)$ will be shaded.

Would you like to know more details about the solution?

Here are 5 related questions for further practice:

1. How would the solution change if the inequality were strict, $x^2 + 6x > -8$?
2. How can you solve a quadratic inequality where the quadratic cannot be factored easily?
3. What would the graph look like if the inequality were $x^2 + 6x \leq -8$?
4. What is the general approach to solving inequalities involving quadratics?
5. Can you solve and graph $x^2 - 4x - 12 \geq 0$?

Tip: When solving inequalities, always pay close attention to whether the inequality is strict ($<$ or $>$) or inclusive ($\leq$ or $\geq$), as it affects the critical points on the graph.