Solve Complex Expression: ((-2)³ + 2⁴)² ÷ 4 + 2³ × 3 ÷ 20

Let's simplify this expression while following the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and that parentheses come before all of these,

Therefore, we'll start by simplifying the expressions in parentheses, note that in this expression there are two pairs of parentheses with a division operation between them, additionally note that inside the left parentheses there is another pair of parentheses with an exponent, so we'll start by simplifying the expression within the inner parentheses that are inside the left parentheses:

$\big(\big((-2)^3+2^4\big)^2:4+2^3\cdot3\big):(4\cdot5)= \\ \big(\big(-8+16\big)^2:4+2^3\cdot3\big):(4\cdot5)= \\ \big(8^2:4+2^3\cdot3\big):(4\cdot5)\\$We simplified the expression in the inner parentheses on the left, this was done in two steps because there was an addition operation between two terms with exponents, therefore, according to the order of operations mentioned above, we first calculated the numerical values of the terms with exponents, this was done while remembering that raising an odd number to a power maintains the sign of the number being raised, then we performed the addition operation within the (inner) parentheses,

Let's continue, for good order, we'll simplify the expression in the left parentheses first and only then simplify the expression in the right parentheses, let's remember again the order of operations mentioned above, therefore we'll start by calculating the terms with exponents since exponents come before multiplication and division, then we'll perform the division and multiplication operations within these parentheses and finally we'll perform the addition operation within the parentheses:

$\big(8^2:4+2^3\cdot3\big):(4\cdot5)=\\ \big(64:4+8\cdot3\big):(4\cdot5)=\\ \big(64:4+8\cdot3\big):(4\cdot5)=\\ \big(16+24\big):(4\cdot5)=\\ 40:20=\\ 2$In the final stages we performed the multiplication within the right parentheses and finally performed the division operation, note that there was no prevention from the first stage to calculate the result of the multiplication in the right parentheses, which we carried through the entire simplification until this stage, however as mentioned before, for good order we preferred to do this in the final stage,

Let's summarize the stages of simplifying the given expression:

$\big(\big((-2)^3+2^4\big)^2:4+2^3\cdot3\big):(4\cdot5)= \\ \big(8^2:4+2^3\cdot3\big):(4\cdot5)=\\ \big(64:4+8\cdot3\big):(4\cdot5)=\\ 40:20=\\ 2$Therefore the correct answer is answer C.