Think back to the long division of your grammar school days, say $\,1839\,$ divided by $\,7\,$:
$\color{red}{\bf{2}}$  $\color{blue}{\bf{6}}$  $\color{purple}{\bf{2}}$  RECALL YOUR THOUGHT PROCESS:  
$7$  $1$  $8$  $3$  $9$  $7$ doesn't go into $1$; it goes into $18$ two times  
$$  $1$  $4$  $2$ times $7$ is $14$; subtract from previous row  
$4$  $3$  $9$  $8  4 = 4$; bring down remaining digits; $7$ goes into $43$ six times  
$$  $4$  $2$  $6$ times $7$ is $42$; subtract from previous row  
$1$  $9$  $3  2 = 1$; bring down remaining digit; $7$ goes into $19$ two times  
$$  $1$  $4$  $2$ times $7$ is $14$; subtract from previous row  
$5$  $9  4 = 5$ is less than the divisor (which is $7$), so stop the process and summarize results 
How many times does
$\,7\,$
go into
$\,1839\,$?
$\,262\,$ times
How much is left over?
$\,5\,$
Notice that we stopped the process with the difference of
$\,5\,$ .
Thus, we never let
$\,7\,$
‘go into’
$\,5\,$ .
So,
$\,5\,$
still needs to be divided by
$\,7\,$
!
So, here is one way to summarize the division problem:
$$\frac{1839}{7} = 262 + \frac{5}{7}\qquad\qquad\qquad (\dagger)$$
Note: the symbol ‘$\dagger$’ is read as ‘dagger’.
You will often see this dagger symbol used to label important information.
Equation $(\dagger)$ often appears in a different form, as shown below:
$$\frac{1839}{7} = 262 + \frac{5}{7}$$  start with equation $(\dagger)$ 
$$7\left( \frac{1839}{7} \right) = 7(262) + 7(\frac 57)$$  multiply both sides by $\,7\,$ 
$\,1839\,$ $=$ $\,7\,$ $\cdot$ $\,262\,$ $+$ $\,5\,$  cancel extra factors of $\,1\,$ This is an equivalent way to summarize the division problem. 
The long division of polynomials process is similar, with just a few extra considerations.
To illustrate, consider this division problem: $\,\displaystyle\frac{x^3  8x + 2}{x+3}\,$
We want to know:
To begin:
$\color{red}{x^2}$  $\color{blue}{\quad 3x}$  $\color{purple}{+\quad 1}$  THE THOUGHT PROCESS:  
$x$  $+\quad 3$  $x^3$  $+$  $0x^2$  $$  $8x$  $+$  $2$  $\,x\,$ goes into $\,x^3\,$ how many times? $\displaystyle\frac{x^3}{x} = $ $\color{red}{x^2}$  
$\color{green}{\bf{}}$  $\color{green}{\bf{(}}$  $x^3$  $+$  $3x^2$  $\color{green}{\bf{)}}$ 
$x^2$
$(x+3)$
$=$
$x^3 + 3x^2\ $; subtract from previous row (put parentheses and minus sign)


$3x^2$  $$  $8x$  $+$  $2$ 
$0x^2  3x^2 = 3x^2$;
bring down remaining terms
;
$x$
goes into
$\,3x^2\,$
how many times?
$\displaystyle\frac{3x^2}{x} = $
$\color{blue}{3x}$


$\color{green}{\bf{}}$  $\color{green}{\bf{(}}$  $3x^2$  $$  $9x$  $\color{green}{\bf{)}}$ 
$3x(x+3) = 3x^2  9x\ $;
subtract from previous row (put parentheses and minus sign)


$x$  $+$  $2$ 
$8x  (9x) = x\ $;
bring down remaining term;
$x\,$
goes into
$x\,$
how many times?
$\displaystyle\frac{x}{x} = $
$\color{purple}{1}$


$\color{green}{\bf{}}$  $\color{green}{\bf{(}}$  $x$  $+$  $3$  $\color{green}{\bf{)}}$ 
$1(x+3) = x + 3\ $;
subtract from previous row (put parentheses and minus sign)


$1$ 
$2  3 = 1\ $;
the degree of
$\,1\,$
is less than the degree of
$\,x+3\,$,
so stop the process and summarize results 
How many times does
$\,x+3\,$
go into
$\,x^3  8x + 2\,$?
$\,x^2  3x + 1\,$ times
How much is left over?
$\,1\,$
Thus:
Of course,
WolframAlpha can do division of polynomials!
The widget below computes $\displaystyle\frac{\text{numerator}}{\text{denominator}}$ .
Type in any numerator and denominator that you want, and then press ‘Submit’.
Have fun!
(comments will appear below) 

On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
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