Consider $-11-23n-2n^{2}$. Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
$$-2n^{2}-23n-11$$
Factor the expression by grouping. First, the expression needs to be rewritten as $-2n^{2}+an+bn-11$. To find $a$ and $b$, set up a system to be solved.
$$a+b=-23$$ $$ab=-2\left(-11\right)=22$$
Since $ab$ is positive, $a$ and $b$ have the same sign. Since $a+b$ is negative, $a$ and $b$ are both negative. List all such integer pairs that give product $22$.
$$-1,-22$$ $$-2,-11$$
Calculate the sum for each pair.
$$-1-22=-23$$ $$-2-11=-13$$
The solution is the pair that gives sum $-23$.
$$a=-1$$ $$b=-22$$
Rewrite $-2n^{2}-23n-11$ as $\left(-2n^{2}-n\right)+\left(-22n-11\right)$.
$$\left(-2n^{2}-n\right)+\left(-22n-11\right)$$
Factor out $-n$ in the first and $-11$ in the second group.
$$-n\left(2n+1\right)-11\left(2n+1\right)$$
Factor out common term $2n+1$ by using distributive property.
$$\left(2n+1\right)\left(-n-11\right)$$
Rewrite the complete factored expression.
$$3\left(2n+1\right)\left(-n-11\right)$$
Steps Using the Quadratic Formula
Quadratic polynomial can be factored using the transformation $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$, where $x_{1}$ and $x_{2}$ are the solutions of the quadratic equation $ax^{2}+bx+c=0$.
$$-6n^{2}-69n-33=0$$
All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. The quadratic formula gives two solutions, one when $±$ is addition and one when it is subtraction.
Now solve the equation $n=\frac{69±63}{-12}$ when $±$ is plus. Add $69$ to $63$.
$$n=\frac{132}{-12}$$
Divide $132$ by $-12$.
$$n=-11$$
Now solve the equation $n=\frac{69±63}{-12}$ when $±$ is minus. Subtract $63$ from $69$.
$$n=\frac{6}{-12}$$
Reduce the fraction $\frac{6}{-12}$ to lowest terms by extracting and canceling out $6$.
$$n=-\frac{1}{2}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $-11$ for $x_{1}$ and $-\frac{1}{2}$ for $x_{2}$.
