Answer: The correct option is (B) [tex]x^2-2x+4.[/tex]
Step-by-step explanation: We are given to find the following quotient:
[tex]Q=(x^3+8)\div(x+2)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
We know the following algebraic formula:
[tex]x^3+a^3=(x+a)(x^2-ax+a^2).[/tex]
So, we have
[tex]x^3+8\\\\=x^3+2^3\\\\=(x+2)(x^2-x\times2+2^2)\\\\=(x+2)(x^2-2x+4).[/tex]
Therefore, from equation (i), we have
[tex]Q\\\\=(x^3+8)\div(x+2)\\\\\\=\dfrac{x^3+8}{x+2}\\\\\\=\dfrac{(x+2)(x^2-2x+4)}{(x+2)}\\\\=x^2-2x+4.[/tex]
Thus, the required quotient is [tex]x^2-2x+4.[/tex]
Option (B) is correct.
