The problem involves simplifying the following expression:

$\left( \frac{3^2}{(-1)^3 + (-3)^3} \right) : \left( 2^2 \cdot (-2)^3 - (-2)^2 \right)$

Step-by-step solution:

1. Evaluate the left-hand side: $\frac{3^2}{(-1)^3 + (-3)^3}$

   • $3^2 = 9$
   • $(-1)^3 = -1$
   • $(-3)^3 = -27$

   Now, evaluate the denominator: $(-1) + (-27) = -28$

   So, the left-hand side becomes: $\frac{9}{-28} = \frac{-9}{28}$

2. Evaluate the right-hand side: $2^2 \cdot (-2)^3 - (-2)^2$

   • $2^2 = 4$
   • $(-2)^3 = -8$
   • $(-2)^2 = 4$

   Now, calculate: $4 \cdot (-8) = -32$ So, the right-hand side becomes: $-32 - 4 = -36$

3. Combine the two expressions: Now, divide the left-hand side by the right-hand side: $\frac{-9}{28} \div (-36) = \frac{-9}{28} \times \frac{1}{-36} = \frac{-9}{28 \times -36} = \frac{-9}{-1008} = \frac{9}{1008}$

4. Simplify the fraction: Simplify $\frac{9}{1008}$: $\frac{9}{1008} = \frac{3}{336} = \frac{1}{112}$

Thus, the result is $\frac{1}{112}$, but since none of the options match this value, it is possible that there was a calculation error or simplification step we might need to reassess. Let's verify the choices again and confirm if any mistake occurred. Do you want more help analyzing this, or could you clarify anything?