The image shows a triangle where two sides are labeled as $3x + 2$ and $7x - 4$. The expression for the perimeter of the triangle is given as $14x + 3$. We need to find the length of the third side of the triangle.

Step-by-step solution:

1. The perimeter of a triangle is the sum of the lengths of all three sides.
2. Let the third side be denoted by $S$.
3. According to the problem, the sum of all three sides equals the perimeter $14x + 3$: $(3x + 2) + (7x - 4) + S = 14x + 3$
4. Combine the like terms from the known side lengths: $3x + 7x + 2 - 4 + S = 14x + 3$ Simplifying: $10x - 2 + S = 14x + 3$
5. Solve for $S$ by isolating it on one side: $S = 14x + 3 - 10x + 2$ Simplifying: $S = 4x + 5$

Final answer:

The length of the third side is $4x + 5$.

Would you like more details or clarification on any part of this? Here are five related questions for further exploration:

1. How would you solve for $x$ if given specific values for the perimeter?
2. Can you verify the result by plugging in values for $x$?
3. What would happen if the perimeter expression was different, say $15x + 6$?
4. How does the triangle's perimeter change if the length of the third side was altered?
5. Could you solve for the triangle's area if it was a right triangle?

Tip: When dealing with multiple variables in an equation, always combine like terms to simplify the problem.