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In algebra, simplifying expressions is a fundamental technique for making equations and problems more manageable. It involves combining like terms, reducing expressions to their simplest form, and applying the order of operations to ensure accuracy. For example, consider the expression \( -5x^2 - 12x^2 + 3x^2 \). Simplifying this focuses on first identifying and then combining the like terms.

Like terms are terms that have the same variables raised to the same power. In our example, all terms are multiples of \( x^2 \) and hence can be combined. Once the like terms are identified, it's a matter of performing arithmetic: \( -5 - 12 + 3 \) in this case. The result is a simplified expression, \( -14x^2 \), which is easier to work with in further calculations or when solving equations.

Understanding like terms in algebra is crucial in the simplification process. Like terms have identical variable parts, which means they have the same variables raised to the same powers. Numbers without variables, known as constant terms, can also be combined with each other.

For instance, in the expression \( 2a^3 + 4a^2b - 3a^3 + 6b \), the terms \( 2a^3 \) and \( -3a^3 \) can be combined because they are both proportional to \( a^3 \) — they are like terms. However, \( 4a^2b \) and \( 6b \) cannot be combined with these because the variable parts differ. It's imperative to only combine terms with matching variable parts to ensure the integrity of the expression.

The order of operations is a set of rules that determines the sequence in which the operations within an expression are performed. Remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction), it helps avoid ambiguity in mathematical expressions.

When simplifying the given expression \( -5x^2 - 12x^2 + 3x^2 \), addition and subtraction are the last steps since there are no parentheses, exponents (outside of the like terms), or multiplication/division to deal with. It's essential to respect this sequence to achieve the correct result. Here, since all terms are like terms, they are combined directly, resulting in the simplified form \( -14x^2 \).