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Understanding 'like terms' in algebra is essential for simplifying expressions and solving equations. Like terms are terms within an algebraic expression that have identical variable parts. This means they have the same variables raised to the same power. For example, in the terms 2x^2 and -5x^2, both have the variable x raised to the power of 2, making them like terms.

When identifying like terms, it's important to note that coefficients, or the numerical factors, do not have to be the same. Only the variable portion must match. So, 3xy and -7xy are like terms because the variable part xy is the same, despite different coefficients.

In the exercise given, we see 4x^2 paired with -5x^2, indicating that they're like terms because of the x^2 part. Recognizing like terms lays the foundation for combining them, which is the next step in simplifying expressions.

Combining like terms refers to the process of adding or subtracting their coefficients while keeping the variable part unchanged. This step is crucial for simplifying algebraic expressions and solving algebraic equations efficiently.

To combine like terms:

• First, identify terms that have the same variable part.
• Then, add or subtract their numerical coefficients.

For instance, combining 4x^2 with -5x^2 simply involves calculating 4 - 5 to get -1, and the resulting term is -x^2. Similarly, -2x and x (which is the same as 1x) can be combined by calculating -2 + 1 to get -x.

In our specific exercise, the student would find the terms with x^2, terms with x, and the constant terms, then combine their coefficients respectively to reach the simplified expression.

Simplifying algebraic expressions is a process that involves several steps to reduce the expression to its simplest form. It's not just aesthetically pleasing, but also makes further algebraic manipulations and solving easier. This process often involves:

• Identifying and combining like terms, as previously discussed.
• Applying the order of operations, which in algebra includes dealing with parentheses, exponents, multiplication, division, addition, and subtraction (PEMDAS).
• Reducing any fractions to their simplest form.
• Factoring, when necessary.

Returning to the exercise, (4x^2 - 2x - 9) + (x - 7 - 5x^2), we begin simplifying by combining like terms. The consolidation of the x^2 terms, x terms, and constants, as illustrated in steps 1 and 2, results in the simplified expression -x^2 - x - 16. This simpler version is beneficial for further mathematical operations and understanding the relationships between variables within the given expression.