What is the quotient of (x³ + 6x² + 11x + 6) ÷ (x² + 4x + 3)?

Step-by-step solution

1. 1

   Predict the degree of the quotient before dividing. Degrees subtract: a degree-3 dividend over a degree-2 divisor gives a quotient of degree $3 - 2 = 1$. So the answer is a linear expression $x + c$, which immediately rules out any constant-only or quadratic option.

2. 2

   First round: divide by the leading term of the divisor. With a quadratic divisor you divide by $x^2$, not $x$:

   $\frac{x^3}{x^2} = x$

   Multiply the whole divisor by $x$ and subtract:

   $x(x^2 + 4x + 3) = x^3 + 4x^2 + 3x$

   $(x^3 + 6x^2 + 11x + 6) - (x^3 + 4x^2 + 3x) = 2x^2 + 8x + 6$

3. 3

   Second round: the degrees cancel exactly. The leftover leads with $2x^2$, the same degree as the divisor:

   $\frac{2x^2}{x^2} = 2$

   So the next quotient term is the constant $2$ — not $2x$. This is the step where mixing up the divisor degree produces a wrong answer.

4. 4

   Subtract and confirm the division terminates. Multiply back:

   $2(x^2 + 4x + 3) = 2x^2 + 8x + 6$

   $(2x^2 + 8x + 6) - (2x^2 + 8x + 6) = 0$

   The remainder is $0$, so the quotient is exactly $x + 2$.

5. 5

   Cross-check by factoring both polynomials. Both factor completely over the integers:

   $x^3 + 6x^2 + 11x + 6 = (x+1)(x+2)(x+3), \qquad x^2 + 4x + 3 = (x+1)(x+3)$

   Cancelling the shared factors $(x+1)(x+3)$ leaves $x + 2$, matching the long division and confirming the division is exact for all $x \neq -1, -3$.