- Identify terms of the same degree (quadratic, linear, constant).
- Combine coefficients of like terms: \((-2+7)x^2 = 5x^2\), \((8+4)x = 12x\), \((-9+2) = -7\).
- State the final simplified polynomial: \(5x^2 + 12x - 7\).
- Conclude the answer: \(\boxed{5x^2 + 12x - 7}\).
Explanation
- Grouping Like Terms
To simplify the expression \(-2x^2 + 8x - 9 + 4x + 7x^2 + 2\), we group the terms based on their degree: the quadratic terms (with \(x^2\)), the linear terms (with \(x\)), and the constant terms.
The quadratic terms are \(-2x^2\) and \(7x^2\).
The linear terms are \(8x\) and \(4x\).
The constant terms are \(-9\) and \(2\).
- Combining Like Terms
Now, we add the coefficients of the terms in each group:
- Quadratic: \(-2x^2 + 7x^2 = (-2 + 7)x^2 = 5x^2\)
- Linear: \(8x + 4x = (8 + 4)x = 12x\)
- Constants: \(-9 + 2 = -7\)
Combining these results, we get the simplified polynomial: \(5x^2 + 12x - 7\).
- Final Result
Comparing our simplified expression \(5x^2 + 12x - 7\) with the given options, we identify that it matches the final choice. Thus, the correct equivalent expression is \(5x^2 + 12x - 7\). The final answer is \(\boxed{5x^2 + 12x - 7}\).
Examples
Simplifying polynomials is a foundational skill in algebra used for tasks like calculating the area of shapes represented by algebraic expressions, managing project budgets where costs are represented as functions, or balancing physical systems. For example, if you are designing a storage box and have separate expressions for the area of the base and the walls, adding them together involves grouping and simplifying these types of terms to find the total surface area.