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The problem states that two sides of a triangle measure \(x^2 + 2x\), and the perimeter of the triangle is \(4x^2 + x\). We need to find the measure of the third side.

Let's call the third side \(S_3\).

The perimeter of a triangle is the sum of the lengths of all three sides. Therefore, we can write the equation for the perimeter as:

\[
P = \text{Side 1} + \text{Side 2} + \text{Side 3}
\]

Substituting the given values:

\[
4x^2 + x = (x^2 + 2x) + (x^2 + 2x) + S_3
\]

Simplifying the terms:

\[
4x^2 + x = 2(x^2 + 2x) + S_3
\]

\[
4x^2 + x = 2x^2 + 4x + S_3
\]

Now, isolate \(S_3\) by subtracting \(2x^2 + 4x\) from both sides:

\[
S_3 = 4x^2 + x - 2x^2 - 4x
\]

Simplify:

\[
S_3 = (4x^2 - 2x^2) + (x - 4x)
\]

\[
S_3 = 2x^2 - 3x
\]

Thus, the measure of the third side is:

\[
S_3 = 2x^2 - 3x
\]

Let me know if you need any additional details!

Here are five related questions:
1. How would the problem change if one of the sides was a constant instead of a quadratic expression?
2. What happens to the third side if \(x\) is negative?
3. Can we determine the type of triangle (scalene, isosceles, etc.) from the given information?
4. How would the solution change if the perimeter was given as a different polynomial expression?
5. Could we solve this problem graphically by plotting the sides?

**Tip**: When solving for unknowns in geometric problems, carefully label and organize all known quantities to avoid confusion!

The measure of two sides of a triangle is x^2 + 2x. If the perimeter is 4x^2 + x, what is the measure of the third side?

Learn how to determine the third side of a triangle when two sides are given by quadratic expressions, and the perimeter is a polynomial. This example includes algebraic concepts, polynomial simplification, and is suitable for students in Grades 8-10.

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