To simplify the equation \( 2(R-1)(R^3-1) = (R^2-1)^2 \), let's follow these steps:

### Step 1: Expand \( (R^2 - 1)^2 \)
We start by expanding the right-hand side of the equation:
\[
(R^2 - 1)^2 = (R^2 - 1)(R^2 - 1)
\]
Use the distributive property (FOIL):
\[
(R^2 - 1)(R^2 - 1) = R^4 - R^2 - R^2 + 1 = R^4 - 2R^2 + 1
\]
So the equation becomes:
\[
2(R - 1)(R^3 - 1) = R^4 - 2R^2 + 1
\]

### Step 2: Factor \( (R^3 - 1) \)
We use the difference of cubes formula:
\[
R^3 - 1 = (R - 1)(R^2 + R + 1)
\]
So, the left-hand side becomes:
\[
2(R - 1)(R - 1)(R^2 + R + 1)
\]
This can be rewritten as:
\[
2(R - 1)^2(R^2 + R + 1)
\]

### Step 3: Combine the equation
Now, we have the simplified form:
\[
2(R - 1)^2(R^2 + R + 1) = R^4 - 2R^2 + 1
\]

This is the simplified form of the equation. To proceed further, one would typically expand the left-hand side and compare terms with the right-hand side.

Would you like me to expand the left-hand side and further verify the equality? 

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**5 Related Questions:**
1. How do you apply the difference of cubes formula in other cases?
2. What are other types of factoring used in algebraic equations?
3. How does one simplify powers of binomials?
4. Can you explain how to solve polynomial equations?
5. What is the importance of verifying identities in algebra?

**Tip:** When simplifying algebraic expressions, always check for common factorizations like difference of squares or cubes.

Explore the detailed solution to simplify the algebraic equation 2(R-1)(R³-1) = (R²-1)². Learn key concepts like the difference of cubes, binomial expansion, and factoring. Suitable for students in Grades 10-12.

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