Factorizing the expression:

To factorize the given expression, we need to simplify it by applying the rules of order of operations and combining like terms. Let's break down the expression step by step:

Step 1: Simplify the exponentiation inside the parentheses:
(2x + 5y)^3 = (2x + 5y)(2x + 5y)(2x + 5y)

Step 2: Expand the cube of a binomial:
(2x + 5y)(2x + 5y)(2x + 5y) = (2x * 2x * 2x) + (2x * 2x * 5y) + (2x * 5y * 2x) + (2x * 5y * 5y) + (5y * 2x * 2x) + (5y * 2x * 5y) + (5y * 5y * 2x) + (5y * 5y * 5y)

Simplifying the above expression gives:
8x^3 + 20x^2y + 20x^2y + 50xy^2 + 20x^2y + 50xy^2 + 50xy^2 + 125y^3
= 8x^3 + 60x^2y + 150xy^2 + 125y^3

Step 3: Simplify the exponentiation of the fractions:
(-5y ÷ 3) ^ 3 = (-5y ÷ 3) * (-5y ÷ 3) * (-5y ÷ 3) = (-5y * -5y * -5y) ÷ (3 * 3 * 3) = -125y^3 ÷ 27

(3z ÷ 4) ^ 3 = (3z ÷ 4) * (3z ÷ 4) * (3z ÷ 4) = (3z * 3z * 3z) ÷ (4 * 4 * 4) = 27z^3 ÷ 64

(-3z ÷ 4 - 2x ÷ 3) ^ 3 = (-3z ÷ 4 - 2x ÷ 3) * (-3z ÷ 4 - 2x ÷ 3) * (-3z ÷ 4 - 2x ÷ 3)

Step 4: Expand the cube of a binomial:
(-3z ÷ 4 - 2x ÷ 3) * (-3z ÷ 4 - 2x ÷ 3) * (-3z ÷ 4 - 2x ÷ 3)
= (-3z ÷ 4) * (-3z ÷ 4) * (-3z ÷ 4) + (-3z ÷ 4) * (-3z ÷ 4) * (-2x ÷ 3) + (-3z ÷ 4) * (-2x ÷ 3) * (-3z ÷ 4)