To solve for the perimeter of the rectangle with given adjacent sides, follow these steps:
1. Identify the expressions for the sides:
We have two adjacent sides of the rectangle given by the expressions:
- Side 1: [tex]\( -6 p^3 + 7 p^2 q^2 + p q \)[/tex]
- Side 2: [tex]\( 7 p q - 5 p^3 + 9 p^2 q^2 \)[/tex]
2. Write down the general formula for perimeter:
For a rectangle, the perimeter [tex]\( P \)[/tex] is given by:
[tex]\[
P = 2(l + w)
\][/tex]
where [tex]\( l \)[/tex] and [tex]\( w \)[/tex] are the lengths of the adjacent sides. In this case, [tex]\( l \)[/tex] is [tex]\( -6 p^3 + 7 p^2 q^2 + p q \)[/tex] and [tex]\( w \)[/tex] is [tex]\( 7 p q - 5 p^3 + 9 p^2 q^2 \)[/tex].
3. Add the expressions for the sides:
Sum the expressions for the side lengths:
[tex]\[
\text{sum of sides} = (-6 p^3 + 7 p^2 q^2 + p q) + (7 p q - 5 p^3 + 9 p^2 q^2)
\][/tex]
4. Simplify the sum of the sides:
Combine like terms:
[tex]\[
\text{sum of sides} = -6 p^3 + 7 p^2 q^2 + p q + 7 p q - 5 p^3 + 9 p^2 q^2
\][/tex]
Combine the [tex]\( p^3 \)[/tex] terms:
[tex]\[
-6 p^3 - 5 p^3 = -11 p^3
\][/tex]
Combine the [tex]\( p^2 q^2 \)[/tex] terms:
[tex]\[
7 p^2 q^2 + 9 p^2 q^2 = 16 p^2 q^2
\][/tex]
Combine the [tex]\( pq \)[/tex] terms:
[tex]\[
p q + 7 p q = 8 p q
\][/tex]
Therefore, the sum of the sides is:
[tex]\[
\text{sum of sides} = -11 p^3 + 16 p^2 q^2 + 8 p q
\][/tex]
5. Calculate the perimeter:
The perimeter is twice the sum of the sides:
[tex]\[
P = 2 \times (-11 p^3 + 16 p^2 q^2 + 8 p q)
\][/tex]
Distribute the 2:
[tex]\[
P = -22 p^3 + 32 p^2 q^2 + 16 p q
\][/tex]
Hence, the perimeter of the rectangle is:
[tex]\[
-22 p^3 + 32 p^2 q^2 + 16 p q
\][/tex]
