To find the perimeter of a rectangle, we use the formula:
[tex]\[
\text{Perimeter} = 2 \times (\text{Sum of the adjacent sides})
\][/tex]
Here, the adjacent sides are given by the expressions:
[tex]\[
\text{Side 1} = -6p^3 + 7p^2q^2 + pq
\][/tex]
[tex]\[
\text{Side 2} = 7pq - 5p^3 + 9p^2q^2
\][/tex]
First, we need to find the sum of the adjacent sides:
[tex]\[
\text{Sum of the adjacent sides} = (\text{Side 1}) + (\text{Side 2})
\][/tex]
Let’s add the given expressions for the adjacent sides:
[tex]\[
\text{Sum} = (-6p^3 + 7p^2q^2 + pq) + (7pq - 5p^3 + 9p^2q^2)
\][/tex]
Combine like terms in the expression:
[tex]\[
\text{Sum} = (-6p^3 - 5p^3) + (7p^2q^2 + 9p^2q^2) + (pq + 7pq)
\][/tex]
Simplify each grouping:
[tex]\[
\text{Sum} = -11p^3 + 16p^2q^2 + 8pq
\][/tex]
Therefore, the sum of the adjacent sides is:
[tex]\[
-11p^3 + 16p^2q^2 + 8pq
\][/tex]
Next, to determine the perimeter, we multiply this sum by 2:
[tex]\[
\text{Perimeter} = 2 \times (-11p^3 + 16p^2q^2 + 8pq)
\][/tex]
Distribute the 2:
[tex]\[
\text{Perimeter} = -22p^3 + 32p^2q^2 + 16pq
\][/tex]
Hence, the perimeter of the rectangle is:
[tex]\[
-22p^3 + 32p^2q^2 + 16pq
\][/tex]
