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When simplifying algebraic expressions, understanding the distributive property is essential. This property states that multiplying a sum or difference by a number is the same as multiplying each term within the sum or difference by that number and then adding or subtracting the products. For instance:
\[ a(b + c) = ab + ac \]
and
\[ a(b - c) = ab - ac \].
In the exercise
\[3(4x - 5)\],
we apply the distributive property to multiply 3 by each term inside the parentheses. The goal is to eliminate parentheses and make the expression easier to work with. The distributive property is a rule that applies regardless of the complexity of the expressions within the parentheses, making it a powerful tool in algebra.

Algebraic simplification involves reducing expressions to their simplest form, making them easier to interpret and solve. It's not just about making it look 'nicer'; it's about creating a form that is operationally effective for further computations or for solving equations. Simplification often involves applying properties such as the distributive property, combining like terms, and reducing fractions.
For our exercise
\[3(4x - 5)\],
the simplification process starts with distribution, as shown in the previous section. Following that, we multiply to get
\[12x - 15\],
which is the simplified form of the expression. Remember, simplification can involve several steps, and each step must be executed correctly to ensure the final solution is accurate. Being systematic and careful at each stage helps avoid errors, ensuring that the expression is correctly simplified.

Multiplication in algebra, particularly when dealing with variables, follows the same basic principles as multiplication of numbers. However, when variables are involved, it adds an additional layer of abstraction. In multiplication, variables are treated as factors that can be multiplied by numbers and other variables.
In the context of our exercise, let's look at how multiplication plays a role in simplifying
\[3(4x - 5)\].
The multiplication
\[3 * 4x\]
results in
\[12x\],
and
\[3 * 5\]
results in
\[15\].
Thus, the multiplication step enables us to combine the numerical coefficient with the variable. It's important to remember that in algebra, coefficients (the numbers in front of variables) are multiplied with their respective variables, while the variables' exponents remain unchanged unless the multiplication involves similar variables.