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!