The image shows the inequality:

$(x^2 - 1)(x^2 - 4) \leq 0$

To solve this inequality, let's break it down into steps:

Step 1: Factorize the expressions

Both $x^2 - 1$ and $x^2 - 4$ are difference of squares, so we can factor them:

$x^2 - 1 = (x - 1)(x + 1)$ $x^2 - 4 = (x - 2)(x + 2)$

So the inequality becomes:

$(x - 1)(x + 1)(x - 2)(x + 2) \leq 0$

Step 2: Determine the critical points

The critical points, where the expression equals zero, occur at: $x = -2, -1, 1, 2$

Step 3: Analyze the intervals

The critical points divide the number line into five intervals:

1. $(-\infty, -2)$
2. $(-2, -1)$
3. $(-1, 1)$
4. $(1, 2)$
5. $(2, \infty)$

We need to check the sign of the product $(x - 1)(x + 1)(x - 2)(x + 2)$ within each interval.

Step 4: Check the sign in each interval

• Interval $(- \infty, -2)$: All four factors are negative, so the product is positive.
• Interval $(-2, -1)$: $x + 2$ is positive, and the others are negative, so the product is negative.
• Interval $(-1, 1)$: $x + 2$ and $x + 1$ are positive, $x - 1$ and $x - 2$ are negative, so the product is positive.
• Interval $(1, 2)$: $x - 2$ is negative, and the others are positive, so the product is negative.
• Interval $(2, \infty)$: All four factors are positive, so the product is positive.

Step 5: Identify where the product is less than or equal to zero

The product is negative or zero in the intervals $(-2, -1)$ and $(1, 2)$, including the critical points $x = -2, -1, 1, 2$ where the product equals zero.

Final Answer:

The solution to the inequality is:

[ [-2, -1] \cup [1, 2] ]

Would you like any further details or have any questions?

Here are 5 questions that relate to this problem:

1. How can inequalities with higher-degree polynomials be solved?
2. What is the significance of the difference of squares in factorization?
3. How do you determine the sign of a polynomial over an interval?
4. How would the solution change if the inequality were strict (i.e., $<$ instead of $\leq$)?
5. What are some common mistakes when solving polynomial inequalities?

Tip: When solving polynomial inequalities, always check the sign of the product within each interval created by the critical points.