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To accurately evaluate algebraic expressions, it's essential to follow the correct order of operations, which is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

This rule dictates the sequence in which you should perform mathematical operations. The parentheses have the highest priority, followed by exponents (like squares or cubes). Afterward, multiplication and division are carried out from left to right, and finally, addition and subtraction are processed, also from left to right.

Let's apply this to our example, using the order of operations to evaluate \( -x^2 - 4 \) when \( x = -2 \). First, we handle the exponent \( (-2)^2 \), and afterwards, subtract 4. The exponents come before the subtraction, which leads to the correct solution. Disregarding this order could yield an incorrect result, emphasizing the importance of PEMDAS in mathematical problem-solving.

Substituting a given value into an algebraic expression is a foundational process in algebra. It involves replacing the variable with its corresponding numerical value. The key to successful substitution is to ensure that you replace every instance of the variable with the given value and be particularly careful with negative numbers.

For our exercise, the value \( x = -2 \) is substituted into the expression, so every instance of \( x \) is replaced with \( -2 \). It's crucial to place parentheses around the substituted value to avoid sign errors, especially when dealing with exponents and negative numbers, just as in \( -(-2)^2 - 4 \). This practice makes the subsequent steps clearer and helps to avoid common mistakes.

Simplifying expressions is the procedure of transforming a complex expression into a simpler or more streamlined form without changing its value. This often involves combining like terms, reducing fractions, or performing arithmetic calculations.

In our case, we simplify the expression after we've followed the order of operations and substituted the value correctly. For the expression \( -(-2)^2 - 4 \), after evaluating the exponent, we then simplify by performing the subtraction, \( -4 - 4 = -8 \). This final step provides us with the simplest form of the expression, representing the value of the entire algebraic expression for the given variable value. Remember, simplifying is all about making the expression as manageable as possible while retaining its original meaning.