The formula for the least common multiple (LCM) of two functions $f(x)$ and $g(x)$ is:

$\text{LCM}(f(x), g(x)) = \frac{f(x) \cdot g(x)}{\text{GCD}(f(x), g(x))}$

You're given that the greatest common divisor (GCD) of $f(x)$ and $g(x)$ is:

$\text{GCD}(f(x), g(x)) = x^2 + x - 2$

This means that we can express the LCM of $f(x)$ and $g(x)$ in terms of $f(x)$ and $g(x)$, with the GCD already provided.

Now, let’s analyze this step-by-step:

Step 1: Recall the LCM formula

The formula for the LCM of two functions is:

$\text{LCM}(f(x), g(x)) = \frac{f(x) \cdot g(x)}{\text{GCD}(f(x), g(x))}$

Step 2: Substitute the GCD into the formula

You are given that the GCD is $x^2 + x - 2$. Substituting this into the formula:

$\text{LCM}(f(x), g(x)) = \frac{f(x) \cdot g(x)}{x^2 + x - 2}$

Step 3: Interpretation

Without knowing the explicit forms of $f(x)$ and $g(x)$, this is the general expression for the LCM of the two functions.

To proceed further, we would need either the forms of $f(x)$ and $g(x)$, or additional information that relates these functions.

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Do you have any specific forms for $f(x)$ and $g(x)$ or any further questions on this problem?

Here are some follow-up questions for exploration:

1. How do we find the LCM and GCD of polynomials?
2. What is the significance of factoring the quadratic expression $x^2 + x - 2$ in this context?
3. Can you provide an example with specific functions for $f(x)$ and $g(x)$?
4. What is the relationship between the LCM and GCD for integers versus polynomials?
5. How can we use the LCM and GCD formulas in real-world problems, such as in signal processing?

Tip: When working with LCM and GCD of polynomials, factoring them completely can often make the process much easier.