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To understand factoring, you first need to know about the Greatest Common Factor (GCF). The GCF of two or more terms is the largest factor that divides all the terms without leaving a remainder.

Finding the GCF involves few steps:

• For numerical coefficients, find the highest number that divides all coefficients.
• For variables, choose the one with the smallest power that is common in all terms.

In our exercise, we have: -9a^5 and -9a^3.

Both terms have a common numerical factor of -9. They also both include the variable 'a' with powers 5 and 3 respectively.
The smallest power is 3, so 'a^3' is the GCF for the variable part.
By combining these, we find the GCF is -9a^3.

Polynomial factorization is the process of expressing a polynomial as a product of its factors.

Factoring polynomials simplifies them and makes solving equations more manageable.

When factoring, your goal is to break down the polynomial into simpler terms (factors) that, when multiplied together, give you the original polynomial.

For instance, in our example: -9a^5 - 9a^3, we identified that -9a^3 is common in both terms.
By factoring -9a^3 out from each term, we get:
-9a^3(a^2 + 1).
Here, -9a^3 and (a^2 + 1) are the factors of the original polynomial.

Factorization helps in simplifying complex polynomials and makes it easier to work with them in solving algebraic expressions.

An algebraic expression involves numbers, variables, and arithmetic operations.
These can be simple, like: 2x + 3, or more complex, like: 4x^2 - 6xy + 9y^2.
Algebraic expressions are presented as combinations of constants and variables linked by operations such as addition, subtraction, multiplication, and division.
In the exercise, the expression -9a^5 - 9a^3 is an algebraic expression composed of two terms.
Factoring out the GCF is one method used to simplify such expressions.

It's crucial to understand how to manipulate algebraic expressions to solve equations and inequalities.
Polynomial factorization, simplifying expressions, and solving equations are essential skills in algebra which build the foundation for advanced mathematics.