To solve the given problem, we need to find the perimeter of the triangle shown in the image. The perimeter of a triangle is the sum of the lengths of its three sides. The side lengths are expressed as polynomials:

1. $4x^2 + 3$
2. $4x^2 + 3x + 5$
3. $2x^2 + 5x + 2$

Step 1: Write the Perimeter as a Sum

To find the perimeter, we add the three polynomials:

$(4x^2 + 3) + (4x^2 + 3x + 5) + (2x^2 + 5x + 2)$

Step 2: Combine Like Terms

Now, let's combine the like terms from all three polynomials:

• $x^2$ Terms:

  • $4x^2 + 4x^2 + 2x^2 = 10x^2$

• $x$ Terms:

  • $3x + 5x = 8x$

• Constant Terms:

  • $3 + 5 + 2 = 10$

Step 3: Write the Result

The perimeter of the triangle, expressed as a polynomial in descending powers of $x$, is:

$10x^2 + 8x + 10$

Final Answer

The perimeter of the triangle is:

$10x^2 + 8x + 10$

Would you like any more details or further explanation on this problem?

Here are some related questions that could help deepen your understanding:

1. How do you add polynomials with different numbers of terms?
2. What are like terms, and why is it important to combine them?
3. How can we represent other geometric properties (like area) using polynomials?
4. How would the solution change if the coefficients had negative values?
5. How do you find the perimeter of other polygons when their sides are given as polynomials?

Tip: When combining like terms in polynomials, always group similar terms (like powers of $x$) before adding to simplify the process!