Let's solve the equation step by step:

Given:
\[
15x^4 - 15x^3 + 15x - 15 = 0
\]

### Step 1: Factor out the common factor
Each term has a common factor of \( 15 \), so we factor it out:
\[
15(x^4 - x^3 + x - 1) = 0
\]

Now, divide both sides of the equation by 15:
\[
x^4 - x^3 + x - 1 = 0
\]

### Step 2: Try factoring the quartic expression
We will attempt to factor the quartic equation \( x^4 - x^3 + x - 1 \). First, let's see if we can use synthetic division or factor by grouping. 

To check for rational roots, use the Rational Root Theorem. The possible rational roots are \( \pm 1 \).

### Step 3: Check possible roots
Substitute \( x = 1 \) into the equation:
\[
1^4 - 1^3 + 1 - 1 = 1 - 1 + 1 - 1 = 0
\]
So, \( x = 1 \) is a root.

### Step 4: Factor \( x - 1 \) from the quartic
Now, we divide \( x^4 - x^3 + x - 1 \) by \( x - 1 \) using synthetic division or polynomial division:

After performing synthetic division, we get:
\[
x^4 - x^3 + x - 1 = (x - 1)(x^3 + x^2 + 1)
\]

### Step 5: Solve the remaining cubic equation
We now solve:
\[
x^3 + x^2 + 1 = 0
\]

This cubic equation doesn't have obvious rational roots, so it would require either numerical methods or deeper analysis for solutions. However, for real roots, it can be shown that there are no real solutions to this equation.

### Final Answer:
Thus, the only real solution is:
\[
x = 1
\]

Would you like details on solving the cubic equation or any further clarification?

### Related Questions:
1. How do you perform synthetic division?
2. What is the Rational Root Theorem and how is it applied?
3. How do you solve a cubic equation analytically?
4. How can you verify the factorization of higher-degree polynomials?
5. What are numerical methods to solve cubic equations?

**Tip:** Factoring out the greatest common factor (GCF) can simplify the equation and make it easier to solve.

Learn how to solve the quartic equation 15x^4 - 15x^3 + 15x - 15 with a detailed step-by-step solution. This problem involves polynomial equations, synthetic division, and application of the Rational Root Theorem. Suitable for advanced high school students.

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