To solve this problem, we need to find the possible length and width of a rectangle whose area is given by the expression $x^2 + 4x$. The area $A$ of a rectangle is calculated as the product of its length $L$ and width $W$:

$A = L \times W$

Step 1: Factor the Area Expression

The area expression is $x^2 + 4x$. We can factor this expression to find possible values for $L$ and $W$:

$x^2 + 4x = x(x + 4)$

This factorization suggests that the length and width could be $x$ and $x + 4$, respectively.

Step 2: Verify Against Answer Choices

Let's check which option matches this factorization:

1. Option 1: Length: $x$, Width: $x + 4$
2. Option 2: Length: $x$, Width: $x^2 + 4$
3. Option 3: Length: $x^2$, Width: $4x$
4. Option 4: Length: $x$, Width: $4x$

Only Option 1 (Length: $x$; Width: $x + 4$) correctly matches the factorization of $x^2 + 4x$.

Conclusion

The correct answer is:

Option 1: Length: $x$; Width: $x + 4$.

Would you like a more detailed explanation or have any questions about the solution?

Here are some additional related questions you might find helpful:

1. How would the answer change if the area was $x^2 - 4x$?
2. What are other ways to find the factors of a quadratic expression like $x^2 + 4x$?
3. How do we handle cases where the area of a rectangle is a polynomial that doesn’t factor neatly?
4. What if the area was $x^2 + 5x + 6$? How would we find possible length and width values?
5. How can we verify if the length and width values are positive, given different values of $x$?

Tip: When dealing with polynomials in geometry problems, factoring can often reveal useful dimensions that fit physical interpretations like length and width.