Let's simplify the expression step by step:

Given the expression:
\[ 8y^2 \div (4y^3 - 4y^2 + 8y) \]

### Step 1: Factor the denominator
Let's factor out the greatest common factor (GCF) in the denominator:

\[
4y^3 - 4y^2 + 8y = 4y(y^2 - y + 2)
\]

So, the expression becomes:

\[
\frac{8y^2}{4y(y^2 - y + 2)}
\]

### Step 2: Simplify the fraction
Now, simplify the fraction by canceling out the common factor of \(4y\) from the numerator and the denominator:

\[
\frac{8y^2}{4y(y^2 - y + 2)} = \frac{2y}{y^2 - y + 2}
\]

So, the simplified form of the expression is:

\[
\boxed{\frac{2y}{y^2 - y + 2}}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you factor polynomials with higher degrees?
2. What is the process for simplifying rational expressions?
3. How do you find the GCF in polynomial expressions?
4. How do you perform long division on polynomials?
5. What are common mistakes to avoid when simplifying expressions?

**Tip:** Always look for common factors before simplifying any algebraic expression to make the process easier.

Learn how to simplify the rational expression 8y^2 / (4y^3 - 4y^2 + 8y) step-by-step. This problem involves algebraic concepts such as rational expressions and factoring. Suitable for high school students in Grades 10-12.

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