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The given expression is:

\[
\frac{4x^2 + 4x}{x^4 + x^3}
\]

### Step 1: Factor the numerator and the denominator.

- **Numerator:** \( 4x^2 + 4x \)
  - Factor out \( 4x \):
    \[
    4x(x + 1)
    \]

- **Denominator:** \( x^4 + x^3 \)
  - Factor out \( x^3 \):
    \[
    x^3(x + 1)
    \]

### Step 2: Simplify the expression.

Now the expression looks like this:

\[
\frac{4x(x + 1)}{x^3(x + 1)}
\]

Since \( (x + 1) \) appears in both the numerator and the denominator, you can cancel \( (x + 1) \) (as long as \( x \neq -1 \)):

\[
\frac{4x}{x^3}
\]

### Step 3: Simplify further.

Now simplify the remaining expression:

\[
\frac{4x}{x^3} = \frac{4}{x^2}
\]

So, the simplified expression is:

\[
\frac{4}{x^2}
\]

### Final Answer:
The completely simplified expression is \( \frac{4}{x^2} \).

Do you have any questions or want more details on any of the steps?

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Here are 5 related questions to explore further:

1. How would the result change if the expression was \( \frac{4x^2 + 2x}{x^4 + x^3} \)?
2. What happens if \( x = -1 \)? Can the expression still be simplified?
3. Can you apply the same factoring technique to other polynomial expressions?
4. How do you simplify expressions with higher-degree polynomials?
5. What other common factorization patterns are useful in simplifying rational expressions?

**Tip:** Always check for common factors in both the numerator and the denominator when simplifying rational expressions.

Simplify the expression: (4x^2 + 4x) / (x^4 + x^3)

Learn how to simplify the rational expression (4x^2 + 4x) / (x^4 + x^3) by factoring both the numerator and denominator, and then cancelling out common factors. This problem covers core algebraic concepts and is suitable for students in Grades 8-10.

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