Solution:
A. Division
\(=\frac{x^2+2 x}{x+1}\\\\=\frac{x(x+1)+x}{x+1}\\\\=\frac{x(x+1)}{x+1}+\frac{x}{x+1}\\\\= x+\frac{x}{x+1}\)
→→→True
B. Multiplication
L HS
\((3 x^4+x^3)(-2 x^4+x^3)=3 x^4\times(-2 x^4+x^3)+x^3 \times (-2 x^4+x^3)\\\\=3 \times -2\times x^4 \times x^4 +3 \times 1\times x^4 \times x^3 + x^3 \times x^4 \times -2+x^3 \times x^3\\\\=-6 x^8+3 x^7-2 x^7+x^6\\\\=-6 x^8+ x^7+x^6\)
R HS
\(=-6x^6+ x^7+x^6\)
LHS ≠ RHS
→→→→False
C: Addition
L HS
\(=(3x^4+x^3)+(-2x^4+x^3)\\\\=3x^4-2x^4+x^3+x^3\\\\ x^4+2 x^3=RHS\)
→→→True
D: Multiplication
LHS
\(=(x^2+2 x)(x+1)\\\\= x^2\times(x+1)+2 x\times (x+1)\\\\= x^3+x^2+2 x^2+2 x\\\\= x^3+3x^2+2 x=RHS\)
→→→True
Option B: is not correctly closed under Multiplication.
