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complete the statement for the rational expression \\(\frac{x^3 + 6x^2 + 11x + 6}{x + 2}\\).
use factoring techniques to easily manipulate the expression.
use the keyboard to type your answer in the box.
\after applying factoring techniques on the given expression, the quotient is \\(square\\) and the remainder is \\(square\\).\

Explanation:

Step1: Factor numerator by grouping

First, group terms in the numerator:
$x^3+6x^2+11x+6 = (x^3+6x^2)+(11x+6)$
Factor common terms from each group:
$= x^2(x+6) + 1(11x+6)$
Instead, use polynomial division or recognize $x=-2$ is a root (since $(-2)^3+6(-2)^2+11(-2)+6 = -8+24-22+6=0$), so $(x+2)$ is a factor. Perform polynomial long division of $x^3+6x^2+11x+6$ by $x+2$.

Step2: Polynomial division setup

Divide $x^3$ by $x$ to get $x^2$. Multiply $(x+2)$ by $x^2$: $x^3+2x^2$. Subtract from numerator:
$(x^3+6x^2+11x+6)-(x^3+2x^2) = 4x^2+11x+6$

Step3: Continue polynomial division

Divide $4x^2$ by $x$ to get $4x$. Multiply $(x+2)$ by $4x$: $4x^2+8x$. Subtract:
$(4x^2+11x+6)-(4x^2+8x) = 3x+6$

Step4: Final division step

Divide $3x$ by $x$ to get $3$. Multiply $(x+2)$ by $3$: $3x+6$. Subtract:
$(3x+6)-(3x+6) = 0$

Answer:

After applying factoring techniques on the given expression, the quotient is $x^2+4x+3$ and the remainder is $0$.