Given the function $F(x) = 2x^2 - 3x + 1$, and the value of $x$ is substituted by $(a+2)$, the goal is to find $F(a+2)$.

Step-by-Step Solution:

1. Substitute $x = a+2$ into the function $F(x)$:

$F(a+2) = 2(a+2)^2 - 3(a+2) + 1$

2. Expand the square $(a+2)^2$:

$(a+2)^2 = a^2 + 4a + 4$

Now substitute this into the function:

$F(a+2) = 2(a^2 + 4a + 4) - 3(a+2) + 1$

3. Distribute the constants:

$F(a+2) = 2a^2 + 8a + 8 - 3a - 6 + 1$

4. Simplify the expression:

$F(a+2) = 2a^2 + (8a - 3a) + (8 - 6 + 1)$ $F(a+2) = 2a^2 + 5a + 3$

Final Answer:

$F(a+2) = 2a^2 + 5a + 3$

Would you like more details on any of the steps?

Here are some related questions you might find useful:

1. How do you complete the square for quadratic functions?
2. What are the properties of quadratic functions and how do you find their vertex?
3. Can you explain the process of expanding binomials like $(a+2)^2$?
4. How can we solve quadratic equations like $2a^2 + 5a + 3 = 0$?
5. How do you graph quadratic functions?

Tip: Always check for common factors after simplifying expressions, as this can sometimes help reduce the complexity further.