Simplify the following and leave the answer in the simplest form: \( \frac{\frac{2015^{45}-2020^{3} \cdot 2015^{5}-2015}{(2020-5)^{2}}}{\frac{1}{2}} \)

To simplify the expression \( \frac{\frac{2015^{45}-2020^{3} \cdot 2015^{5}-2015}{(2020-5)^{2}}}{\frac{1}{2}} \), we start by rewriting it by multiplying by the reciprocal: \[ = 2 \cdot \frac{2015^{45}-2020^{3} \cdot 2015^{5}-2015}{(2020-5)^{2}} \] Now, we calculate \( 2020 - 5 \): \[ 2020 - 5 = 2015 \] Substituting this back into the expression gives us: \[ = 2 \cdot \frac{2015^{45} - 2020^{3} \cdot 2015^{5} - 2015}{2015^{2}} \] Next, let's simplify the numerator: \[ 2015^{45} - 2020^{3} \cdot 2015^{5} - 2015 \] This can be factored by taking out a common factor of \( 2015 \): \[ = 2015 \left( 2015^{44} - 2020^{3} \cdot 2015^{4} - 1 \right) \] Inserting this back into our expression gives: \[ = 2 \cdot \frac{2015 \left( 2015^{44} - 2020^{3} \cdot 2015^{4} - 1 \right)}{2015^{2}} \] We can simplify \( \frac{2015}{2015^{2}} \) to \( \frac{1}{2015} \): \[ = 2 \cdot \frac{1}{2015} \left( 2015^{44} - 2020^{3} \cdot 2015^{4} - 1 \right) \] Thus, the expression simplifies to: \[ = \frac{2 \left( 2015^{44} - 2020^{3} \cdot 2015^{4} - 1 \right)}{2015} \] Now we just need to evaluate \( 2020^3 \): \[ 2020 = 2000 + 20 \] Using the binomial theorem: \[ 2020^3 = (2000 + 20)^3 = 2000^3 + 3 \cdot 2000^2 \cdot 20 + 3 \cdot 2000 \cdot 20^2 + 20^3 \] Calculating each component: \[ = 8 \times 10^{9} + 3 \times 4 \times 10^{6} + 3 \times 2000 \times 400 + 8000 \] Combining these evaluations could be intensive, so we use \( 2020^3 \) directly in: \[ = 2020^3 \cdot 2015^4 \] Thus, our simplified overall expression looks like this: \[ = \frac{2}{2015} \left( 2015^{44} - 2020^{3} \cdot 2015^{4} - 1 \right) \] This is the simplest form considering all calculations. We conclude with the final answer: \[ \frac{2 \left( 2015^{44} - 2020^{3} \cdot 2015^{4} - 1 \right)}{2015} \]