To find the difference of the given polynomials, we need to perform polynomial subtraction step by step.
Given polynomials:
[tex]\[ P_1 = 8r^6s^3 - 9r^5s^4 + 3r^4s^5 \][/tex]
[tex]\[ P_2 = 2r^4s^5 - 5r^3s^6 - 4r^5s^4 \][/tex]
We are asked to find:
[tex]\[ P_1 - P_2 = (8r^6s^3 - 9r^5s^4 + 3r^4s^5) - (2r^4s^5 - 5r^3s^6 - 4r^5s^4) \][/tex]
First, let's distribute the negative sign through [tex]\( P_2 \)[/tex]:
[tex]\[ P_1 - P_2 = 8r^6s^3 - 9r^5s^4 + 3r^4s^5 - 2r^4s^5 + 5r^3s^6 + 4r^5s^4 \][/tex]
Next, we combine like terms:
1. Terms with [tex]\( r^6s^3 \)[/tex]:
[tex]\[ 8r^6s^3 \][/tex]
2. Terms with [tex]\( r^5s^4 \)[/tex]:
[tex]\[ -9r^5s^4 + 4r^5s^4 = -5r^5s^4 \][/tex]
3. Terms with [tex]\( r^4s^5 \)[/tex]:
[tex]\[ 3r^4s^5 - 2r^4s^5 = r^4s^5 \][/tex]
4. Terms with [tex]\( r^3s^6 \)[/tex]:
[tex]\[ 5r^3s^6 \][/tex]
Therefore, the difference of the polynomials is:
[tex]\[ 8r^6s^3 - 5r^5s^4 + r^4s^5 + 5r^3s^6 \][/tex]
Matching this with the given options, we find that the correct answer is:
[tex]\[ \boxed{8 r^6 s^3 - 5 r^5 s^4 + r^4 s^5 + 5 r^3 s^6} \][/tex]
