# How do you long divide  (x^2+4x+4) / (x+2) ?

Jul 16, 2015

It is a challenge to format and there are various ways of writing the details. Here's the one that I learned and teach (using a standard textbook in the US):

#### Explanation:

To divide $\frac{{x}^{2} + 4 x + 4}{x + 2}$

Write the divisor, $x + 2$ on the left. Then the dividend, ${x}^{2} + 4 x + 4$ under the division thing. (I don't know its name.)

$\text{ " " }$$- - - - -$
x+2 ) ${x}^{2}$ $+ 4 x$ $+ 4$

Look at the first terms on the divisor $x + 2$ and the dividend ${x}^{2} + 4 x + 4$. What do we need to multiply $x$ by, to get ${x}^{2}$? We need to multiply by $x$. Write that $x$ on the top line:

$\text{ " " " " " } x$
$\text{ " " }$$- - - - -$
x+2 ) ${x}^{2}$ $+ 4 x$ $+ 4$

Now multiply $x$ times the divisor, $x + 2$, to get ${x}^{2} + 2 x$ and write that under the dividend.

$\text{ " " " " " }$ $x$
$\text{ " " }$$- - - - -$
x+2 ) ${x}^{2}$$\text{ }$ $+ 4 x$$\text{ }$ $+ 4$
$\text{ " " " "" }$ ${x}^{2}$ $\text{ }$$+ 2 x$
$\text{ " " }$$- - - - -$

Now we need to subtract ${x}^{2} + 2 x$ from the dividend. (You may find it simpler to change the signs and add.)

$\text{ " " " " " }$ $x$
$\text{ " " }$$- - - - -$
x+2 ) ${x}^{2}$ $\text{ }$$+ 4 x$ $\text{ }$$+ 4$
$\text{ " " }$ $\textcolor{red}{-} {x}^{2} \textcolor{red}{-} 2 x$
$\text{ " " }$$- - - - -$
$\text{ "" "" "" "" }$ $2 x$$\text{ }$ $+ 4$

Now, look at the first terms of the divisor and the expression on the last line. What do we need to multiply $x$ (the first term of the divisor) by to get $2 x$ (the first term on the last line)? We need to multiply by $2$
Write $+ 2$ on the top line, then multiply $2$ times the divisor $x + 2$, to get $2 x + 4$ and write it underneath.

$\text{ " " " " " }$ $x$$\text{ }$ $+ 2$
$\text{ " " }$$- - - - -$
x+2 ) ${x}^{2}$ $\text{ }$$+ 4 x$ $\text{ }$$+ 4$
$\text{ " " }$ $\textcolor{red}{-} {x}^{2} \textcolor{red}{-} 2 x$
$\text{ " " }$$- - - - - -$
$\text{ " " " " "" }$ $2 x$$\text{ }$ $+ 4$
$\text{ " " " " " " }$ $2 x$$\text{ }$ $+ 4$
$\text{ " " }$$- - - - - -$

Now subtract (change the signs and add), to get:

$\text{ " " " " " }$ $x$$\text{ }$ $+ 2$
$\text{ " " }$$- - - - -$
x+2 ) ${x}^{2}$ $\text{ }$$+ 4 x$ $\text{ }$$+ 4$
$\text{ " " }$ $\textcolor{red}{-} {x}^{2} \textcolor{red}{-} 2 x$
$\text{ " " }$$- - - - - -$
$\text{ " " " " "" }$ $2 x$$\text{ }$ $+ 4$
$\text{ " " " " }$ $\textcolor{red}{-} 2 x$$\text{ }$ $\textcolor{red}{-} 4$
$\text{ " " }$$- - - - - -$
$\text{ " " " " " " " " "" " " }$ $0$

The quotient is $x + 2$ and the remainder is $0$.