Answer: The correct option is (C) [tex]x^2+2x+3.[/tex]
Step-by-step explanation: We are given to find the quotient by dividing the following :
[tex]Q=\dfrac{x^3+3x^2+5x+3}{x+1}.[/tex]
We will try to factorize the numerator ad then divide by the denominator to find the quotient.
We have
[tex]Q\\\\\\=\dfrac{x^3+3x^2+5x+3}{x+1}\\\\\\=\dfrac{x^2(x+1)+2x(x+1)+3(x+1)}{x+1}\\\\\\=\dfrac{(x+1)(x^2+2x+3)}{(x+1)}\\\\=x^2+2x+3.[/tex]
Thus, the required quotient is [tex]x^2+2x+3.[/tex]
Option (C) is correct.
