https://file.aiquickdraw.com/mathbot/file/1728264350400vjn8qr83.jpg

To find the perimeter of the triangle shown in the image, we need to add up the lengths of all three sides. The side lengths are expressed as algebraic expressions:

1. One side is \( 2x - 3 \),
2. Another side is \( 2x^2 + 8x \),
3. The third side is \( 3x^2 - 6x + 5 \).

The perimeter, \( P \), is simply the sum of these three expressions:

\[
P = (2x - 3) + (2x^2 + 8x) + (3x^2 - 6x + 5)
\]

Now, combine like terms:

1. Combine the \( x^2 \) terms: \( 2x^2 + 3x^2 = 5x^2 \),
2. Combine the \( x \) terms: \( 2x + 8x - 6x = 4x \),
3. Combine the constant terms: \( -3 + 5 = 2 \).

So the perimeter expression is:

\[
P = 5x^2 + 4x + 2
\]

This is the expression for the perimeter of the triangle.

Would you like more details on any step, or do you have any questions?

Here are 5 follow-up questions to expand on this:

1. What happens to the perimeter if \( x = 2 \)?
2. How can you interpret the perimeter if \( x = 0 \)?
3. What is the degree of the polynomial that represents the perimeter?
4. How would you solve for \( x \) if a specific perimeter value is given?
5. How does the coefficient of \( x^2 \) affect the shape of the triangle?

**Tip:** When adding algebraic expressions, always group and combine like terms to simplify the equation efficiently.

Add to find the perimeter of the triangle with sides: 2x - 3, 2x^2 + 8x, and 3x^2 - 6x + 5.

Learn how to find the perimeter of a triangle with sides expressed as polynomial functions: 2x - 3, 2x^2 + 8x, and 3x^2 - 6x + 5. This problem covers algebraic concepts and polynomial addition, suitable for students in Grades 7-9.

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