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When we talk about the 'distribution of variables' in algebra, we're referring to a technique often known as the distributive property. This is an essential tool when simplifying algebraic expressions. Let's say you have a variable outside the parentheses, like in the expression \( -2r(r+s) \). What you want to do is multiply that outer variable by each term inside the parentheses.

The process looks like this: first, you multiply \( -2r \) by \( r \), yielding \( -2r^2 \). Then, you distribute \( -2r \) to \( s \) which gives \( -2rs \). Putting it all together, you have two instances of \( -2rs \) since both \( r \) and \( s \) are distributed, leading to \( -2r^2 - 2rs - 2rs \). Remember that every term inside the parentheses gets multiplied by the term outside; this is the cornerstone of distribution in algebra.

After distributing variables, the next step involves 'combining like terms,' which is a method for simplifying algebraic expressions further. Terms are 'like' if they have the same variables raised to the same powers. For example, \( -2r^2 \) and \( -4rs \) are not like terms because one is an \( r^2 \) term and the other is an \( rs \) term.

In our exercise, after distributing the variables, we obtained the expression \( -2r^2 - 2rs - 2rs - 2s^2 \). Notice how \( -2rs \) appears twice? These are like terms because they share the same variables with the same exponents. We combine these by adding their coefficients: \( -2 - 2 = -4 \), resulting in \( -4rs \). The simplified version of our expression now reads \( -2r^2 - 4rs - 2s^2 \). This tactic helps significantly reduce and tidy up algebraic expressions, making them easier to work with.

Now, let's delve into 'polynomial simplification.' A polynomial is an expression that includes terms like \( r^2 \) or \( rs \), which can include constants, variables, and exponents. Simplifying a polynomial means performing all possible distribution and combination of like terms to express it in its simplest form.

In our given problem, we started with \( -2 r(r+s)(r+s) \) and after distribution, we reached \( -2r^2 - 4rs - 2s^2 \). There are no further like terms to combine, so we've simplified the polynomial as much as we can. Remember, the order of the terms doesn't matter in addition, so \( -2r^2 - 2s^2 - 4rs \) is equally valid. Whenever you encounter a polynomial, break it down step by step—distribute first, then combine like terms—and you'll arrive at a simplified expression in no time.