To solve the given mathematical expression, we need to follow the order of operations (PEMDAS/BODMAS):

\[ \frac{4^2 \cdot 25^2 + 32 \cdot 125}{2^3 \cdot 5} \]

Let's break it down step by step:

1. Calculate the exponents:
\[ 4^2 = 16 \]
\[ 25^2 = 625 \]
\[ 2^3 = 8 \]

2. Substitute these values back into the expression:
\[ \frac{16 \cdot 625 + 32 \cdot 125}{8 \cdot 5} \]

3. Perform the multiplications:
\[ 16 \cdot 625 = 10000 \]
\[ 32 \cdot 125 = 4000 \]
\[ 8 \cdot 5 = 40 \]

4. Substitute these values back into the expression:
\[ \frac{10000 + 4000}{40} \]

5. Perform the addition in the numerator:
\[ 10000 + 4000 = 14000 \]

6. Finally, divide the result by the denominator:
\[ \frac{14000}{40} = 350 \]

So, the value of the given expression is \( 350 \).