Sure, let's solve the problem step by step.
First, let's identify the expressions for the adjacent sides of the rectangle:
1. The first side of the rectangle is given as:
[tex]\[
\text{side1} = -6p^3 + 7p^2q^2 + pq
\][/tex]
2. The second side of the rectangle is given as:
[tex]\[
\text{side2} = 7pq - 5p^3 + 9p^2q^2
\][/tex]
Next, to find the perimeter of the rectangle, we use the formula for the perimeter [tex]\(P\)[/tex] of a rectangle, which is given by:
[tex]\[
P = 2(\text{side1} + \text{side2})
\][/tex]
Substitute the given expressions for [tex]\(\text{side1}\)[/tex] and [tex]\(\text{side2}\)[/tex] into the formula:
[tex]\[
P = 2 \left( (-6p^3 + 7p^2q^2 + pq) + (7pq - 5p^3 + 9p^2q^2) \right)
\][/tex]
Now, we combine the like terms inside the parentheses:
- Combine the [tex]\(p^3\)[/tex] terms:
[tex]\[
-6p^3 - 5p^3 = -11p^3
\][/tex]
- Combine the [tex]\(p^2q^2\)[/tex] terms:
[tex]\[
7p^2q^2 + 9p^2q^2 = 16p^2q^2
\][/tex]
- Combine the [tex]\(pq\)[/tex] terms:
[tex]\[
pq + 7pq = 8pq
\][/tex]
So, the expression inside the parentheses simplifies to:
[tex]\[
-11p^3 + 16p^2q^2 + 8pq
\][/tex]
Finally, we multiply by 2 to get the perimeter:
[tex]\[
P = 2 \left( -11p^3 + 16p^2q^2 + 8pq \right)
\][/tex]
So, the perimeter [tex]\(P\)[/tex] of the rectangle is:
[tex]\[
P = -22p^3 + 32p^2q^2 + 16pq
\][/tex]
Thus, the perimeter of the rectangle is:
[tex]\[
-22p^3 + 32p^2q^2 + 16pq
\][/tex]
