The given expression is:

$(5 - a)(a^2 - 10 - 3a)$

To expand this expression, we apply the distributive property (also known as FOIL when applied to binomials), where we multiply each term in the first parenthesis by each term in the second parenthesis.

Steps:

1. Distribute $5$ to each term in $a^2 - 10 - 3a$: $5(a^2 - 10 - 3a) = 5a^2 - 50 - 15a$

2. Distribute $-a$ to each term in $a^2 - 10 - 3a$: $-a(a^2 - 10 - 3a) = -a^3 + 10a + 3a^2$

3. Now, combine all terms: $(5a^2 - 50 - 15a) + (-a^3 + 10a + 3a^2)$

4. Simplify by combining like terms: $-a^3 + (5a^2 + 3a^2) + (-15a + 10a) - 50$ $= -a^3 + 8a^2 - 5a - 50$

So, the expanded form of the expression is:

$-a^3 + 8a^2 - 5a - 50$

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

1. How do you factor a cubic polynomial like $-a^3 + 8a^2 - 5a - 50$?
2. What are the properties of cubic polynomials?
3. How do you use synthetic division to solve polynomials?
4. How can you solve for $a$ given a cubic polynomial?
5. What is the significance of finding the roots of a polynomial?

Tip: When expanding, always distribute each term completely and then carefully combine like terms to avoid mistakes.