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Understanding the basic arithmetic operations is the foundation of math. These operations include addition (+), subtraction (-), multiplication (×), and division (÷). In any mathematical expression, it's important to perform these operations in the correct order. This order is often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

In the given exercise, multiplication is the first operation we encounter. Looking at each term: for 1 multiplied by -4, the two integers are multiplied to get (-4). Similarly, 2 multiplied by 5 results in 10. Lastly, 3 multiplied by -6 gives (-18). Notice how we handle positive and negative numbers; a positive times a negative gives a negative result, and vice versa. This attention to signs is crucial in arithmetic operations to ensure the accuracy of the solution.

Evaluating expressions is a fundamental skill in algebra. It involves performing calculations according to the order of operations to find the value of an expression. When you come across a mathematical expression, your task is to simplify it by carrying out the operations correctly.

Our example consisted of the expression: 1(-4) + 2(5) + 3(-6). To evaluate, we started by handling the multiplication within each group. This process involves not just carrying out the operation but also being careful with the signs of the numbers involved. Once we completed the multiplications, we moved on to the final step, which is addition and subtraction. Since there's no specific order between addition and subtraction, we work from left to right, combining the products to reach the final evaluated expression value of -12).

Algebraic problem solving involves a systematic approach to solving mathematical problems that require more than basic arithmetic. After establishing the foundation with arithmetic operations and evaluating expressions, we begin to solve more complex algebraic problems. We use variables, formulate equations, and sometimes manipulate those equations to find unknown values.

In simpler cases like our exercise, it's about orderly executing operations. But as problems become more complex, understanding the principles of algebra, such as combining like terms, understanding the distributive property, and isolating variables, becomes essential. Algebraic problem-solving requires a sequential approach, much like we demonstrated in our step-by-step solution, where we carefully evaluated each term after multiplication and then summed up the results for the final answer.