(x³-2)! - (y²+3)!
———————————— =
(y+2)! + (x²-4)! - (y³-1)!

To solve the given expression:

\[
\frac{(x^3 - 2)! - (y^2 + 3)!}{(y + 2)! + (x^2 - 4)! - (y^3 - 1)!}
\]

we'll need to follow these steps:

### Step 1: Analyze and simplify individual components
Let's rewrite the expression and analyze each factorial term in detail.

- \( (x^3 - 2)! \)
- \( (y^2 + 3)! \)
- \( (y + 2)! \)
- \( (x^2 - 4)! \)
- \( (y^3 - 1)! \)

These are factorial expressions, which can only be computed if the arguments are non-negative integers. Therefore, for this expression to be meaningful, each of the terms inside the factorials must be non-negative integers.

### Step 2: Establish restrictions for valid solutions
For factorials to be defined, we need:
- \( x^3 - 2 \geq 0 \Rightarrow x^3 \geq 2 \Rightarrow x \geq \sqrt[3]{2} \)
- \( y^2 + 3 \geq 0 \Rightarrow y^2 \geq -3 \) (which is always true for any real \( y \))
- \( y + 2 \geq 0 \Rightarrow y \geq -2 \)
- \( x^2 - 4 \geq 0 \Rightarrow x^2 \geq 4 \Rightarrow x \geq 2 \text{ or } x \leq -2 \)
- \( y^3 - 1 \geq 0 \Rightarrow y^3 \geq 1 \Rightarrow y \geq 1 \)

Thus, the constraints for this expression are:
- \( x \geq 2 \)
- \( y \geq 1 \)

### Step 3: Factorial simplification or substitution
At this stage, it's hard to simplify the expression further without specific values for \( x \) and \( y \). The best next step is to substitute numerical values for \( x \) and \( y \) within the valid range (e.g., \( x = 2 \), \( y = 1 \)) and compute the factorials.

Would you like to proceed by choosing specific values for \( x \) and \( y \), or would you prefer further guidance on the general form?

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### Follow-up Questions:
1. Would you like to explore the behavior of the expression with specific values of \( x \) and \( y \)?
2. Do you need more details on how factorials work?
3. Would you like to see an example of factorial expansion for other values of \( x \) and \( y \)?
4. Should we discuss the domain restrictions in more detail?
5. Would you like to explore how changing \( x \) and \( y \) affects the expression?

**Tip:** Remember that factorials are only defined for non-negative integers, so always check the argument inside the factorial for validity.

(x³-2)! - (y²+3)! ———————————— = (y+2)! + (x²-4)! - (y³-1)!

Learn how to solve a complex factorial expression involving polynomials such as (x³-2)! and (y²+3)!. This detailed step-by-step guide explores factorials, polynomial functions, and domain restrictions, suitable for advanced high school students.

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