To write an expression for the perimeter of the triangle, we need to sum up the lengths of its three sides. The side lengths given in terms of $n$ are:

• One side: $2n + 2$
• Another side: $n + 1$
• The third side: $3n + 3$

Step 1: Expression for the Perimeter

The perimeter $P$ of the triangle is the sum of all the sides:

$P = (2n + 2) + (n + 1) + (3n + 3)$

Step 2: Simplify the Expression

Combine like terms:

$P = 2n + n + 3n + 2 + 1 + 3$

$P = (2n + n + 3n) + (2 + 1 + 3)$

$P = 6n + 6$

So, the simplified expression for the perimeter is:

$P = 6n + 6$

Step 3: Evaluate when $n = 4$

Now, substitute $n = 4$ into the simplified expression:

$P = 6(4) + 6 = 24 + 6 = 30$

Thus, the perimeter of the triangle when $n = 4$ is 30.

Do you need further details or explanations? Here are 5 related questions:

1. How do we determine the perimeter for any polygon?
2. Can this formula for the perimeter apply to other types of polygons?
3. How does simplifying an algebraic expression help in real-world problems?
4. What are some geometric properties of triangles with expressions for side lengths?
5. Can you find the perimeter for a different value of $n$?

Tip: Always remember to combine like terms to simplify expressions easily!