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?

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.