How do you factor ${n^2} + 7n - 44$ completely ?

Hint: In this question, we are given a quadratic expression and we have to factorize it. Use splitting the middle term method to find the factors. After finding factors, take out the common numbers or variables and factorize the expression. Finally we get the required answer.

Complete step-by-step solution:
We are given a quadratic expression ${n^2} + 7n - 44$.
Let us find two such factors of $ - 44$, such that when they are added or subtracted, they give us $7$.
If we observe, then two such factors are $4$ and $11$. One of the two factors needs to be negative. But, how to decide which factor should be negative? For this, we will check the coefficient of $n$. If it is negative, the bigger factor out of the two will be negative and if the coefficient is positive, the smaller factor will be taken as negative. Now, since the required coefficient is positive, our factors will be $ - 4$ and $11$.
$ \Rightarrow {n^2} + 11n - 4n - 44$
Taking $n$ common from the first two terms and $4$ from the last two terms,
$ \Rightarrow n\left( {n + 11} \right) - 4\left( {n + 11} \right)$
Making factors,
$ \Rightarrow \left( {n + 11} \right)\left( {n - 4} \right)$

Hence, $(n+11)$ and $(n-4)$ are the factors of ${n^2} + 7n - 44$.

Note: We can also factorize the expression using the formula –
$n = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
But using this formula, we will directly get the values of $n$. Using the values, we will have to find the factors. Let us see how.
We know that $a = 1,b = 7,c = - 44$. Let us put the values in the formula –
$ \Rightarrow n = \dfrac{{ - 7 \pm \sqrt {{7^2} - 4 \times 1 \times \left( { - 44} \right)} }}{2}$
Simplifying,
$ \Rightarrow n = \dfrac{{ - 7 \pm \sqrt {49 + 176} }}{2}$
$ \Rightarrow n = \dfrac{{ - 7 + \sqrt {225} }}{2},\dfrac{{ - 7 - \sqrt {225} }}{2}$
Now, we know that $\sqrt {225} = 15$.
$ \Rightarrow n = \dfrac{{ - 7 + 15}}{2},\dfrac{{ - 7 - 15}}{2}$
We will get,
$ \Rightarrow n = \dfrac{8}{2},\dfrac{{-22}}{2}$
$ \Rightarrow n = 4,-11$
Now, we will make factors using these values.
$ \Rightarrow n - 4 = 0,n + 11 = 0$
$ \Rightarrow \left( {n - 4} \right)\left( {n + 11} \right) = 0$
Hence, the two factors are $\left( {n - 4} \right)\left( {n + 11} \right)$.