The given expression is:

$5x^2y^4 \cdot (-7x^3y)$

Step-by-step simplification:

1. Multiply the constants: $5 \times (-7) = -35$

2. Multiply the powers of $x$: Using the rule $x^a \cdot x^b = x^{a+b}$, we add the exponents of $x$: $x^2 \cdot x^3 = x^{2+3} = x^5$

3. Multiply the powers of $y$: Similarly, apply the rule to the powers of $y$: $y^4 \cdot y^1 = y^{4+1} = y^5$

Final simplified expression:

$-35x^5y^5$

Would you like further clarification or more details on this? Here are some related questions to expand your understanding:

1. How do you apply the distributive property when simplifying expressions?
2. Can you explain the laws of exponents used in multiplication?
3. What happens if the bases are different in an expression like this?
4. How would you simplify if there were division involved instead of multiplication?
5. What is the general rule for simplifying variables with exponents?

Tip: When multiplying terms with exponents, always add the exponents if the bases are the same!