2–3÷[(2*2-5)²*3*2+1]-2+7*2

Work inside the innermost parentheses first (2*2-5), multiplication
comes before subtraction, so multiply 2*2 and get 4, so replace 2*2 
by 4 and copy everything else over:

2–3÷[(4-5)²*3*2+1]-2+7*2

Work inside the innermost parentheses first (4-5), There is only
the one subtraction so we subtract 4-5 getting -1 and we replace
(4-5) by (-1) and copy everything else over:

2–3÷[(-1)²*3*2+1]-2+7*2

Work inside the innermost parentheses first, [(-1)²*3*2+1].
Exponents come before multiplication or addition, so do (-1)²
and get 1, so replace (-1)² by 1 and copy everything else over:

2–3÷[1*3*2+1]-2+7*2

Work inside the innermost parentheses first, [1*3*2+1].
The first multiplication comes first, so do 1*3
and get 3, so replace 1*3 by 3 and copy everything else over:

2–3÷[3*2+1]-2+7*2

Work inside the innermost parentheses first, [3*2+1].
The multiplication comes before addition, so do 3*2
and get 6, so replace 3*2 by 6 and copy everything else over:

2–3÷[6+1]-2+7*2

Work inside the parentheses first [6+1], There is only
the one addition so we add 6+1 getting 7 and we replace
[6+1] by 7 and copy everything else over:

2–3÷7-2+7*2

We do multiplications and divisions as they come left
to right. The division is before the multiplication so 
we do 3÷7, getting .  We replace the 3÷7 by 
 and copy everything else over:

2–-2+7*2

Next we do the multiplication first.  7*2 = 14 so we replace
the 7*2 by 14 and copy the rest over:

2–-2+14

We do additions and subtractions left to right, so we subtract
2- and get  = . So we
replace 2– by  and copy the rest over:

-2+14

We do additions and subtractions left to right, so we subtract
-2 and get  = . So we
replace -2 by  and copy the rest over:

+14

We add those:   and get 

 = 

Edwin