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The substitution method is a fundamental approach in algebra for evaluating functions. This method involves replacing a variable with a specific value. In the given exercise, we need to find F(-2) for the function F(x) = 12 - x + 3x^2.
First, identify the function and then substitute the given value of the variable. Let's replace x with -2 in the expression F(x).
The equation becomes F(-2) = 12 - (-2) + 3(-2)^2. This simplification will allow us to solve for the function at x = -2. The key here is to handle the arithmetic operations carefully, ensuring each substitution step is accurate.

Polynomial functions are expressions involving variables raised to whole number exponents. They might have constants and coefficients. For example, the function in the exercise, F(x) = 12 - x + 3x^2, is a quadratic polynomial because it involves a term with \(x^2\).
Polynomial functions can take on different forms based on their degree (the highest exponent of the variable). In our case, it is a second-degree polynomial because the highest power of x is 2.
Evaluating polynomial functions involves substituting values for the variable and then performing the arithmetic operations to find the result. These operations include addition, subtraction, and multiplication, all of which are straightforward once substitution is done.

Evaluating a function means finding the output value given an input value. Let's follow the steps to evaluate F(-2) for the polynomial F(x) = 12 - x + 3x^2:
1. **Identify the Function:** Given F(x) = 12 - x + 3x^2.
2. **Substitute the Value:** Replace x with -2 in the function, making it F(-2) = 12 - (-2) + 3(-2)^2.
3. **Simplify the Expression:** This step involves simplification:

• First, handle the terms inside the parentheses: -(-2) becomes +2.
• Next, compute the squared term: (-2)^2 = 4.
• Then multiply by the coefficient: 3 x 4 = 12.
• Combine terms: 12 + 2 + 12.

4. **Combine Results:** Finally, add the simplified terms: 14 + 12 = 26. Therefore, F(-2) = 26.
Through these steps, you can systematically find the output of any function given an input.