If $f(x)$ is a polynomial of degree 2, its general form is:

$f(x) = ax^2 + bx + c$

The Taylor series expansion of $f(x)$ around any point $x = a$ is given by:

$f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f^{(3)}(a)}{3!}(x - a)^3 + \cdots$

Since $f(x)$ is a polynomial of degree 2, all higher derivatives of $f(x)$ (i.e., $f^{(n)}(x)$ for $n \geq 3$) will be zero. This means the Taylor series will terminate after the $(x - a)^2$ term.

Thus, the Taylor series expansion will have three terms:

1. $f(a)$
2. $f'(a)(x - a)$
3. $\frac{f''(a)}{2!}(x - a)^2$

Would you like more details or have any questions?

Here are five related questions:

1. What is the Taylor series expansion of a general cubic polynomial?
2. How do we compute the Taylor series expansion for non-polynomial functions like $e^x$?
3. How does the choice of the expansion point $a$ affect the Taylor series?
4. What is the relationship between the Taylor series and the Maclaurin series?
5. How many terms does the Taylor series of a linear function have?

Tip: The higher the degree of the polynomial, the more terms you'll find in its Taylor series expansion.