We have the function:

$y = f(x) = \frac{x^3 + 2x^2 - 8}{x^2 + x - 2}$

Let's analyze this function and explore some key aspects of it. We'll begin by factoring both the numerator and denominator, if possible, and see if we can simplify the expression.

Step 1: Factor the denominator

The denominator is $x^2 + x - 2$. We can factor it:

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

Step 2: Factor the numerator

The numerator is $x^3 + 2x^2 - 8$. We can factor out $x$ from the first two terms:

$x^3 + 2x^2 - 8 = x(x^2 + 2x) - 8$

Notice, we can try factoring $x^2 + 2x - 8$. This gives us:

$x^3 + 2x^2 - 8 = x(x + 4)(x - 2)$

Step 3: Simplify the function

Now we can rewrite the function as:

$f(x) = \frac{x(x + 4)(x - 2)}{(x - 1)(x + 2)}$

This is a simplified form, and it cannot be reduced further unless we factor more from the numerator and denominator. However, notice that the expression is undefined for $x = 1$ and $x = -2$, because the denominator becomes zero at these values.

Summary:

• The function is undefined at $x = 1$ and $x = -2$.
• The simplified expression is:

$f(x) = \frac{x(x + 4)(x - 2)}{(x - 1)(x + 2)}$

Would you like to explore any further aspects of this function, such as finding its domain, range, limits, or behavior at critical points? Let me know! Here's a tip:

Tip: Always check for values that make the denominator zero, as they will indicate where the function is undefined.

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Here are 5 related questions you might find interesting:

1. What is the domain of the function $f(x)$?
2. How do you find the limits of $f(x)$ as $x \to 1$ or $x \to -2$?
3. How do you find the critical points of the function $f(x)$?
4. What is the behavior of the function $f(x)$ as $x \to \infty$?
5. Can you find any horizontal or vertical asymptotes for the function $f(x)$?