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When dealing with fractions, it's important to understand the roles of the numerator and the denominator. The numerator is the number above the fraction line, representing the number of parts we have. The denominator, located below the fraction line, represents the total number of equal parts in a whole.

In the expression \( \frac{3(-8)-(-2)(-4-5)}{(-2)(-3)} \), the numerator is the part of the fraction you simplify first, \( 3(-8)-(-2)(-4-5) \). This part of the expression tells us what we have to compute at the top.

For complex expressions, it helps to handle the numerator and the denominator separately for clarity. In this exercise, solving the numerator gives us \( -6 \).

The denominator is \( (-2)(-3) \), which simplifies to \( 6 \). Understanding how these two components fit together is crucial. Simplifying them step by step leads us to the ultimate solution of the expression.

Parentheses play a crucial role in algebra, as they dictate the order of calculations in expressions. When you see parentheses, solve the expressions within them first. This prioritization avoids any miscalculations that might occur if expressions are solved out of order.

In the given expression, we have several parentheses: \((-2)(-4-5)\). The inner expression \((-4-5)\) must be solved first. This sum equals \(-9\). Then, multiplying \(-2\) by \(-9\) gives us \(18\).

This procedure must be applied consistently. Without it, expressions can easily become misleading or incorrect. Using parentheses correctly ensures that each operation follows the necessary mathematical order to achieve the correct outcome.

Arithmetic operations form the foundation of simplifying algebraic expressions. They include addition, subtraction, multiplication, and division. Each of these operations must be performed in a specific order, often guided by the parentheses present in the expression. This is known as the order of operations.

For the expression \( \frac{3(-8)-(-2)(-4-5)}{(-2)(-3)} \), we perform multiplication inside the parentheses first, handling the expressions \(3(-8)\) and \((-2)(-9)\). The product of these is \(-24\) and \(18\), respectively.

Next, we proceed to perform the subtraction for the entire numerator: \(-24 + 18\), simplifying to \(-6\).

The product in the denominator is \((-2)(-3)\), simplifying to \(6\). Finally, division of the numerator by the denominator, \(\frac{-6}{6}\), simplifies to \(-1\). Each step handled carefully ensures the expression is simplified correctly.