Question 4 Simplify \( \left(x^{3}+6 x^{2}+11 x+6\right) \div(x+3) \) \[ \]

To simplify \( \left(x^{3}+6 x^{2}+11 x+6\right) \div(x+3) \), we can use polynomial long division. Dividing \( x^3+6x^2+11x+6 \) by \( x+3 \), we start by dividing the leading terms: \( x^3 \div x = x^2 \). Multiply \( x^2 \) by \( x + 3 \) to get \( x^3 + 3x^2 \) and subtract this from the original polynomial. We repeat this process, bringing down each term as needed, until we find that \( (x^3 + 6x^2 + 11x + 6) = (x+3)(x^2 + 3x + 2) \), leading to a quotient of \( x^2 + 3x + 2 \), which can be further factored into \( (x+1)(x+2) \). Thus, the final simplified form is: \[ \left(x^{3}+6 x^{2}+11 x+6\right) \div(x+3) = (x + 1)(x + 2) \] This means the simplified form is \( x^2 + 3x + 2 \).