The problem presented is the polynomial inequality:

$(x - 1)^2 \geq 4$

Step 1: Isolate the expression.

Start by solving the corresponding equality to find the critical points:

$(x - 1)^2 = 4$

Take the square root of both sides:

$x - 1 = \pm 2$

Thus, the two solutions are:

$x = 1 + 2 = 3 \quad \text{or} \quad x = 1 - 2 = -1$

So the critical points are $x = -1$ and $x = 3$.

Step 2: Solve the inequality.

Now we solve the inequality:

$(x - 1)^2 \geq 4$

This inequality is satisfied when $(x - 1)^2 = 4$, which we already found as $x = -1$ and $x = 3$. Additionally, since $(x - 1)^2$ represents a parabola opening upwards, it is greater than or equal to 4 outside the interval $(-1, 3)$. So, we test values in the intervals:

1. For $x < -1$ (e.g., $x = -2$): $(-2 - 1)^2 = 9 \geq 4$ This is true, so $x < -1$ satisfies the inequality.

2. For $-1 < x < 3$ (e.g., $x = 0$): $(0 - 1)^2 = 1 \not\geq 4$ This is false, so the inequality does not hold in this interval.

3. For $x > 3$ (e.g., $x = 4$): $(4 - 1)^2 = 9 \geq 4$ This is true, so $x > 3$ satisfies the inequality.

Step 3: Interval notation.

The solution includes $x = -1$ and $x = 3$ because the inequality is "greater than or equal to." Therefore, the solution in interval notation is:

$(-\infty, -1] \cup [3, \infty)$

This is the set of all $x$ values that satisfy the inequality.

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Would you like further details or have any questions?

Here are some related questions to deepen understanding:

1. How do you determine when to use a union of intervals in inequalities?
2. What is the graphical representation of $(x - 1)^2 = 4$?
3. Why do we need to check each interval between critical points for inequalities?
4. What happens if the inequality were strict, $(x - 1)^2 > 4$?
5. How do you solve inequalities with absolute values?

Tip: When solving polynomial inequalities, always solve the related equation first to find critical points and then check the intervals around those points.