The given expression is:

$5m^2n^3 \times 3mn^4$

Step-by-step simplification:

1. Multiply the coefficients:
   $5 \times 3 = 15$

2. Combine the powers of $m$:
   $m^2 \times m = m^{2+1} = m^3$

3. Combine the powers of $n$:
   $n^3 \times n^4 = n^{3+4} = n^7$

Now, combine everything:

$15m^3n^7$

Final simplified expression:

$15m^3n^7$

Would you like more details or have any further questions? Here are some related questions you might find interesting:

1. How do you handle expressions with negative exponents?
2. What is the product rule for exponents?
3. How would you simplify expressions with fractional exponents?
4. What happens when you divide terms with the same base?
5. How do you simplify expressions with different variables?

Tip: When multiplying terms with the same base, just add the exponents.