Answer: The required quotient is [tex](x^2-2x+4).[/tex]
Step-by-step explanation: We are given to find the quotient for the following division :
[tex](x^3+8)~~\textup{divided by}~~(x+2).[/tex]
To find the quotient, first we try to factorize the numerator and check if there is any common term in both the numerator and denominator to cancel.
We will be using the following factorization formula :
[tex]x^3+a^3=(x+a)(x^2-xa+a^2).[/tex]
We have
[tex](x^3+8)~~\textup{divided by}~~(x+2)\\\\\\=\dfrac{x^3+8}{x+2}\\\\\\=\dfrac{x^3+2^3}{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]
