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An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Variables are symbols that stand for unspecified numbers or values. In our example, the algebraic expression is p(m-n)-q(n-m), where p, q, m, and n are variables that represent certain numbers.

Understanding how to manipulate these expressions is crucial in solving algebra problems. Operations such as addition, subtraction, multiplication, and division can be performed on algebraic expressions, just like with numeric terms. The goal is frequently to simplify the expression or solve for a variable. In the given exercise, we simplify the expression by transforming and factoring it, which is an essential skill in algebra.

The distributive property is a cornerstone in algebra. It states that, for any numbers or expressions a, b, and c, the operation a(b + c) is the same as ab + ac. In other words, you can distribute the multiplication of one number over the addition of two others.

In the context of our example, we apply the distributive property in reverse during the factoring process. In Step 3 of the solution, when we see q[-(m-n)], we distribute the negative sign to the expressions inside the parenthesis, which gives us q(m-n), with a positive sign instead of a negative one. This process is a critical part of simplifying and refactoring expressions in algebra.

To factor out common terms means to identify and 'pull out' any factors that appear in each term of an algebraic expression. A factor is a number or expression that divides another number or expression evenly—without a remainder. When we factor out a common term from an expression, we simplify it so that the product of the common factor and the remaining expression represents the original expression.

In Step 4 of the original problem's solution, we see that (m-n) is a common term in both parts of the expression p(m-n) + q(m-n). Factoring it out, we combine the separate terms into a single product (p+q)(m-n). Factoring is often used to simplify expressions, solve equations, and help understand complex algebraic relationships more easily. It's a powerful tool that can make complex algebraic expressions much more manageable.