X^2+6x+8>=0

To solve the inequality \( x^2 + 6x + 8 \geq 0 \), we first find the roots of the corresponding equation \( x^2 + 6x + 8 = 0 \).

We can factor the quadratic:

\[ x^2 + 6x + 8 = (x + 2)(x + 4) \]

Setting the equation to zero gives us the roots:

\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \] \[ x + 4 = 0 \quad \Rightarrow \quad x = -4 \]

The roots of the equation are \( x = -2 \) and \( x = -4 \). Now, we will test the intervals defined by these roots to determine where the quadratic expression is non-negative.

The critical points split the number line into three intervals:

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

We will test a point from each interval:

1. Interval \( (-\infty, -4) \): Test with \( x = -5 \) \[ (-5 + 2)(-5 + 4) = (-3)(-1) = 3 \, (> 0) \] Thus, the expression is positive in this interval.

2. Interval \( (-4, -2) \): Test with \( x = -3 \) \[ (-3 + 2)(-3 + 4) = (-1)(1) = -1 \, (< 0) \] Thus, the expression is negative in this interval.

3. Interval \( (-2, \infty) \): Test with \( x = 0 \) \[ (0 + 2)(0 + 4) = (2)(4) = 8 \, (> 0) \] Thus, the expression is positive in this interval.

Now we consider the critical points where the expression equals zero:

• At \( x = -4 \): \( (x + 2)(x + 4) = 0 \)
• At \( x = -2 \): \( (x + 2)(x + 4) = 0 \)

Since the inequality is \( \geq 0 \), we include the points:

Putting this all together, the solution to the inequality \( x^2 + 6x + 8 \geq 0 \) is:

\[ (-\infty, -4] \cup [-2, \infty) \]