(3r-2t)⁴

I like the way your teacher showed you:

 [(3r-2t)²]²

You need this two step rule for squaring any polynomial:

To square any polynomial, the two-step rule is

The sum of:

1. the squares of every term: 

plus the sum of

2. Twice every product of every pair of terms. 

--------------------------------

Let's do it this way:

 [(3r-2t)²]²

Do the inside (3r-2t)² first by the two-step rule:

The sum of:

1. the squares of every term: 
 (3r)² + (-2t)² = 9r² + 4t² 

plus

2. twice every product of every pair of terms.  
(there's only one pair) 2(3r)(-2t) = -12rt

So [(3r-2t)²]² becomes

[9r² + 4t² - 12rt]²

Use the two-step rule again:

The sum of:

1. the squares of every term: 

(9r²)² + (4t²)² + (-12rt)² = 81r⁴+ 16t⁴+ 144r²t² 

plus the sum of

2. Twice every product of every pair of terms. 

2(9r²)(4t²) + 2(9r²)(-12rt) + 2(4t²)(-12rt) = 

     72r²t² - 216r³t - 96rt³ 

So we have:
  
81r⁴+ 16t⁴+ 144r²t² + 72r²t² - 216r³t - 96rt³

The terms in r²t² combine and so the final answer is

81r⁴+ 16t⁴+ 216r²t² - 216r³t - 96rt³

Edwin