Let's solve the equation:

\[
32535 = 234x(x^3 - 34x^2 + 12x - 48)
\]

### Step 1: Divide both sides by 234 to simplify the equation.

\[
\frac{32535}{234} = x(x^3 - 34x^2 + 12x - 48)
\]

We calculate:

\[
\frac{32535}{234} = 139
\]

So the equation becomes:

\[
139 = x(x^3 - 34x^2 + 12x - 48)
\]

### Step 2: Expand the right-hand side of the equation.

\[
139 = x^4 - 34x^3 + 12x^2 - 48x
\]

### Step 3: Rearrange the equation.

\[
x^4 - 34x^3 + 12x^2 - 48x - 139 = 0
\]

We now have a **quartic equation**: 

\[
x^4 - 34x^3 + 12x^2 - 48x - 139 = 0
\]

### Step 4: Solve the quartic equation.

This equation can be solved using numerical methods such as Newton's method or by approximating the roots through trial and error. Would you like me to compute the roots numerically or use a specific method?

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Would you like details on the solving process, or should I continue with root calculations?

### Related questions:
1. What is a quartic equation, and how is it typically solved?
2. How do you use Newton's method to solve equations like this?
3. Are there analytical solutions for all quartic equations?
4. Can we use factorization for equations of this type?
5. What are some real-world applications of quartic equations?

**Tip:** When solving higher-order polynomials, checking for rational roots first can simplify the process.

Learn how to solve the quartic equation x^4 - 34x^3 + 12x^2 - 48x - 139 = 0 using numerical methods. This problem covers advanced algebraic concepts, including polynomial expansion, rational root theorem, and Newton's method. Suitable for high school students in Grades 10-12.

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