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When dealing with polynomials, one of the foundational skills is combining like terms. This allows us to simplify expressions into a more manageable form. Like terms are terms in a polynomial that have the same variables raised to the same power. For instance, in the expression

• \(3x^2\) and \(-8x^2\) are like terms because they both have the variable \(x\) raised to the second power.
• Similarly, the constants \(-4\) and \(9\) are considered like terms because they do not contain any variables.

To combine these terms, we simply sum up their coefficients. For the \(x^2\) terms: \[3x^2 - 8x^2 = -5x^2\] This subtraction is straightforward as we subtract as if they were simple numbers or integers. The expression simplifies when like terms are combined, making it easier to work with.

Removing parentheses is an essential step in simplifying polynomial expressions. The initial expression \[(3x^2 + 3x - 4) + (-8x^2 + 9)\] uses parentheses to define separate sets of terms. An important aspect of removing parentheses is to watch for the sign in front of them.
In this exercise, there is a plus sign between the two sets:

• The plus sign allows us to remove the parentheses without changing the signs of the terms inside.

The expression transforms smoothly into: \[3x^2 + 3x - 4 - 8x^2 + 9\]No changes are needed for any signs within the parentheses. This leads seamlessly to the next step, where we can combine like terms. Properly removing parentheses sets the stage for accurate calculation.

Simplifying expressions is the final step in cleaning up a polynomial to its most reduced form. After combining like terms and removing parentheses, we take what’s left and combine, if possible, further to ensure the expression is as simple as it can be.
From the expression \[3x^2 + 3x - 4 - 8x^2 + 9\]we identified our like terms and combined them:

• The \(x^2\) terms combined to get \(-5x^2\).
• The constant terms combined to result in \(5\).
• The \(x\) term, being standalone, is simply \(3x\).

Thus, after simplifying, we are left with \[-5x^2 + 3x + 5\]This expression is now in its simplest and most manageable form, ready for further operations or evaluations if needed. Simplification makes it easier for future calculations or when substituting values for variables. Understanding each step ensures clarity and builds a solid foundation for algebraic manipulation.