Okay, let's find the product [tex]\(\left(4 p^3 s^2-4 p^2 s^3\right)\left(3 p^2 s^2+5 p^5 s\right)\)[/tex].
Here is the step-by-step solution:
1. Distribute each term in the first polynomial to each term in the second polynomial.
This means we will expand by multiplying each term in [tex]\((4 p^3 s^2 - 4 p^2 s^3)\)[/tex] by each term in [tex]\((3 p^2 s^2 + 5 p^5 s)\)[/tex].
2. Multiply [tex]\(4 p^3 s^2\)[/tex] by each term in the second polynomial:
- [tex]\(4 p^3 s^2 \cdot 3 p^2 s^2 = 4 \cdot 3 \cdot p^{3+2} \cdot s^{2+2} = 12 p^5 s^4\)[/tex]
- [tex]\(4 p^3 s^2 \cdot 5 p^5 s = 4 \cdot 5 \cdot p^{3+5} \cdot s^{2+1} = 20 p^8 s^3\)[/tex]
3. Multiply [tex]\(-4 p^2 s^3\)[/tex] by each term in the second polynomial:
- [tex]\(-4 p^2 s^3 \cdot 3 p^2 s^2 = -4 \cdot 3 \cdot p^{2+2} \cdot s^{3+2} = -12 p^4 s^5\)[/tex]
- [tex]\(-4 p^2 s^3 \cdot 5 p^5 s = -4 \cdot 5 \cdot p^{2+5} \cdot s^{3+1} = -20 p^7 s^4\)[/tex]
4. Combine all the terms we obtained:
- [tex]\(12 p^5 s^4\)[/tex]
- [tex]\(20 p^8 s^3\)[/tex]
- [tex]\(-12 p^4 s^5\)[/tex]
- [tex]\(-20 p^7 s^4\)[/tex]
5. Write the final expression by putting together all the terms:
So, the product of [tex]\((4 p^3 s^2 - 4 p^2 s^3)\)[/tex] and [tex]\((3 p^2 s^2 + 5 p^5 s)\)[/tex] is:
[tex]\[ 20 p^8 s^3 - 20 p^7 s^4 + 12 p^5 s^4 - 12 p^4 s^5 \][/tex]
This is the expanded form of the given expressions.
