The equation you provided is:

\[
x^2 + x^3 + y^4 - 36 = 9
\]

To simplify and solve this equation, we first rearrange it:

\[
x^2 + x^3 + y^4 - 36 = 9
\]

Subtract 9 from both sides:

\[
x^2 + x^3 + y^4 - 36 - 9 = 0
\]

\[
x^2 + x^3 + y^4 - 45 = 0
\]

This is a non-linear equation with two variables, \(x\) and \(y\). To proceed further, more specific information about \(x\) and \(y\) would be required, such as particular values or constraints for these variables. Without additional constraints, this equation has infinitely many solutions for different combinations of \(x\) and \(y\).

Would you like me to solve for a specific variable, or assume any particular values?

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Here are five follow-up questions to consider:
1. Should we solve for \(x\) or \(y\) if one of the variables is known?
2. Are there any restrictions on \(x\) and \(y\) (e.g., integers or real numbers)?
3. Would you like to graph the function to visualize its behavior?
4. Should the equation be factored or manipulated differently for further simplification?
5. Are you working with a specific mathematical context (e.g., geometry or physics)?

**Tip:** For complex multi-variable equations, plotting the function can help visualize relationships between variables and identify solutions.

Explore the process of solving the non-linear multivariable equation x^2 + x^3 + y^4 - 36 = 9. Learn about algebraic manipulation techniques, the properties of non-linear equations, and methods for handling equations with two variables, x and y. Suitable for students in Grades 10-12.

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