Start out with this:
      
                         `
a - 2 )a³ - 5a² + 13a - 14
     
The "a" goes into the "a²" "a" times, so write "a" over the "- 5a²"

             a²          `      
a - 2 )a³ - 5a² + 13a - 14
       
Now multiply the a² by the -2 and get -2a² and write that below the 
"- 5a²":

             a²          `            
a - 2 )a³ - 5a² + 13a - 14
          - 2a²
                       
                      
Now multiply the a² by the a, getting a³, so write that under the 
a³.                        
   
             a²          `                
a - 2 )a³ - 5a² + 13a - 14
       a² - 2a²
                        
Now draw a line and subtract (-5a²)-(-2a²) = -5a²+2a² = -3a². So write that under
the - 2a² termm, and bring down the next term, " 13a"
                                                     
             a²          ` 
a - 2 )a³ - 5a² + 13a - 14   
       a² - 2a²
          - 3a² + 13a

"a" goes into "-3a²"  "-3a" times. So we write "- 3a" above the "+ 13a" 
     
             a² -  3a    `
a - 2 )a³ - 5a² + 13a - 14
       a² - 2a²
          - 3a² + 13a
           
Now multiply the -3a by both the a and the -2, getting -3a² + 6a.  Write
that at the bottom.
    
             a² -  3a    `
a - 2 )a³ - 5a² + 13a - 14
       a² - 2a²
          - 3a² + 13a
           -3a² +  6a
                   
Draw a line and subtract: (+ 13a) - (+ 6a) = 13a - 6a = 7a. 
So write 7a under the 6a.  Bring down the last term "-14"
                   
             a² -  3a    ` 
a - 2 )a³ - 5a² + 13a - 14
       a² - 2a²
          - 3a² + 13a
           -3a² +  6a
                   7a - 14
                    
"a" goes into "7a" "7" times, so write "+ 7" above the "- 14"
     
             a² -  3a +  7
a - 2 )a³ - 5a² + 13a - 14
       a² - 2a²
          - 3a² + 13a
           -3a² +  6a
                   7a - 14

Multiply the "+ 7" by both the "a" and the "- 2", and get "7a - 14", so
write this at the bottom 

             a² -  3a +  7
a - 2 )a³ - 5a² + 13a - 14
       a² - 2a²
          - 3a² + 13a
           -3a² +  6a
                   7a - 14
                   7a - 14 

Draw a line and subtract: (-14) - (-14) = -14 + 14 = 0
Write 0 at the bottom under the -14

             a² -  3a +  7
a - 2 )a³ - 5a² + 13a - 14
       a² - 2a²
          - 3a² + 13a
           -3a² +  6a
                   7a - 14
                   7a - 14
                         0

Since the remainder is 0, the answer is a² - 3a + 7 that you
see at the top.

Edwin