Let's solve the given problem:

We need to multiply:

$-2u^9 \cdot 7u$

First, simplify by multiplying the coefficients:

$-2 \times 7 = -14$

Now, multiply the variables:

$u^9 \times u^1 = u^{9+1} = u^{10}$

So, the simplified expression is:

$-14u^{10}$

Would you like a more detailed explanation, or do you have any questions?

Here are some related questions to expand your understanding:

1. How do you multiply coefficients in algebraic expressions?
2. What is the rule for multiplying variables with the same base?
3. Can you apply this rule to other exponents, like $a^4 \times a^5$?
4. How would you multiply a negative number by a positive one?
5. What changes if there were multiple variables involved, like $-2u^9 v^3 \times 7uv$?

Tip: When multiplying powers with the same base, simply add the exponents.