(t4 − 2t² + 4t − 5) ÷ (t² − 3)

Put in terms with 0 coefficients for place holders:

(t4 + 0t³ - 2t² + 4t - 5) ÷ (t² + 0t - 3)

           ________________________
t² + 0t - 3)t4 + 0t³ - 2t² + 4t - 5

Divide t4 ÷ t² = t² so put that above the line above the - 3t²


                        t²             
t² + 0t - 3)t4 + 0t³ - 2t² + 4t - 5

Multiply the t² times the t² + 0t - 3, t²(t² + 0t - 3) = t4 + 9t³ - 3t² 
and put that under the corresponding terms and draw a line underneath:

                        t²         
t² + 0t - 3)t4 + 0t³ - 2t² + 4t - 5
            t4 + 0t³ - 3t²
            
Now subtract (t4 + 0t³ - 2t²) - (t4 + 0t³ - 3t²) = 
t4 + 0t³ - 2t² - t4 - 0t³ + 3t² = 0t³ + t². Put that below the line and 
bring down the next term + 4t:

                        t²        
t² + 0t - 3)t4 + 0t³ - 2t² + 4t - 5
            t4 + 0t³ - 3t²
                 0t³ +  t² + 4t

Divide 0t³ ÷ t² = 0t so put that above the line above the + 4t
 
                        t² + 0t    
t² + 0t - 3)t4 + 0t³ - 2t² + 4t - 5
            t4 + 0t³ - 3t²
                 0t³ +  t² + 4t

Multiply the 0t times the t² + 0t - 3, 0t(t² + 0t - 3) = 0t³ + 0t² + 0t
and put that under the corresponding terms and draw a line underneath:

                        t² + 0t    
t² + 0t - 3)t4 + 0t³ - 2t² + 4t - 5
            t4 + 0t³ - 3t²
                 0t³ +  t² + 4t
                 0t³ + 0t² + 0t

Now subtract (0t³ +  t² + 4t) - (0t³ + 0t³ - 3t²) = 
0t³ +  t³ + 4t - 0t³ + 0t² + 0t = t³ + 4t. Put that below the 
line and bring down the next (last) term - 5:

                        t² + 0t    
t² + 0t - 3)t4 + 0t³ - 2t² + 4t - 5
            t4 + 0t³ - 3t²
                 0t³ +  t² + 4t
                 0t³ + 0t² + 0t
                        t² + 4t - 5

Divide t² ÷ t² = 1 so put that above the line above the - 5

                        t² + 0t + 1
t² + 0t - 3)t4 + 0t³ - 2t² + 4t - 5
            t4 + 0t³ - 3t²
                 0t³ +  t² + 4t
                 0t³ + 0t² + 0t
                        t² + 4t - 5

Multiply the 1 times the t² + 0t - 3, 1(t² + 0t - 3) =  t³ + 0t - 3 
and put that under the corresponding terms and draw a line underneath:


                        t² + 0t + 1
t² + 0t - 3)t4 + 0t³ - 2t² + 4t - 5
            t4 + 0t³ - 3t²
                 0t³ +  t² + 4t
                 0t³ + 0t² + 0t
                        t² + 4t - 5
                        t³ + 0t - 3

Now subtract (t² + 4t - 5) - (t² + 0t - 3) = 
              t² + 4t - 5 - t² - 0t² + 3 =  
              4t - 2. Put that below the line

                        t² + 0t + 1
t² + 0t - 3)t4 + 0t³ - 2t² + 4t - 5
            t4 + 0t³ - 3t²
                 0t³ +  t² + 4t
                 0t³ + 0t² + 0t
                        t² + 4t - 5
                        t³ + 0t - 3
                             4t - 2

There are no more terms to bring down, so we are done with
the dividion, and are ready to write the answer:

We use the formula Answer = QUOTIENT + 

                   t² + 0t + 1 + 

 Now we can drop the 0 terms:

                   t² + 1 +  

Edwin