Factor the expression by grouping. First, the expression needs to be rewritten as $20n^{2}+an+bn+2$. To find $a$ and $b$, set up a system to be solved.
$$a+b=13$$ $$ab=20\times 2=40$$
Since $ab$ is positive, $a$ and $b$ have the same sign. Since $a+b$ is positive, $a$ and $b$ are both positive. List all such integer pairs that give product $40$.
$$1,40$$ $$2,20$$ $$4,10$$ $$5,8$$
Calculate the sum for each pair.
$$1+40=41$$ $$2+20=22$$ $$4+10=14$$ $$5+8=13$$
The solution is the pair that gives sum $13$.
$$a=5$$ $$b=8$$
Rewrite $20n^{2}+13n+2$ as $\left(20n^{2}+5n\right)+\left(8n+2\right)$.
$$\left(20n^{2}+5n\right)+\left(8n+2\right)$$
Factor out $5n$ in the first and $2$ in the second group.
$$5n\left(4n+1\right)+2\left(4n+1\right)$$
Factor out common term $4n+1$ by using distributive property.
$$\left(4n+1\right)\left(5n+2\right)$$
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 $ax^{2}+bx+c=0$.
$$20n^{2}+13n+2=0$$
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 $n=\frac{-13±3}{40}$ when $±$ is plus. Add $-13$ to $3$.
$$n=-\frac{10}{40}$$
Reduce the fraction $\frac{-10}{40}$ to lowest terms by extracting and canceling out $10$.
$$n=-\frac{1}{4}$$
Now solve the equation $n=\frac{-13±3}{40}$ when $±$ is minus. Subtract $3$ from $-13$.
$$n=-\frac{16}{40}$$
Reduce the fraction $\frac{-16}{40}$ to lowest terms by extracting and canceling out $8$.
$$n=-\frac{2}{5}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $-\frac{1}{4}$ for $x_{1}$ and $-\frac{2}{5}$ for $x_{2}$.
Multiply $\frac{4n+1}{4}$ times $\frac{5n+2}{5}$ by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
