Is f(x)=2x^{2}+4 a polynomial function?

Explanation:
A polynomial consists of a term or terms, each term has a constant (a number), possibly 1 or -1 (or even 0) times a variable raised to a positive whole number power. We also count a single constant as a term. ( sometimes we say it includes ${\displaystyle x^0}$, but we don't write it).
If the polynomial has just one term, it is called a monomial.
The constant (number) in the term is called the coefficient.
("terms" are thing thar are added together, "factors" are things that are multiplied)
Here are some examples of polynomials (compare to the definition given above)
7x+3 the terms are 7x and 3
${\displaystyle 3x^2-5x+2}$ the terms are ${\displaystyle 3x^2}$, and -5x and 2
The polynomials above have integer coefficient, the next polynomial has rational number coefficients:
${\displaystyle \frac{3}{5}{x}^6-x^5-\frac{1}{10}{x}^2+x-9}$ the terms are ${\displaystyle \frac{3}{5}{x}^6,-x^5,-\frac{1}{10}{x}^2}$, x, and (the constant term) - 9 If we wanted to, we could also list ${\displaystyle 0x^4}$, and ${\displaystyle 0x^3}$ (and even ${\displaystyle 0x^7}$) and so on).
Notice: (1) we do not usually write terms that have 0 coeffiicient, although we could. And in some situations we put them in while we work on a problem, then take them out at the end.
(2) we usually do not write a coefficient that is 1 or -1.
The following two examples are not polynomials:
${\displaystyle \frac{7}{x^3}+3x+4}$ is not a polynomial -- polynomials do not have the variable in the denominator.
${\displaystyle 3\sqrt{x}+2}$. is not a polynomial -- polynomials do not have roots of the variable.
But ${\displaystyle \sqrt{3}{x}^2-4x+\sqrt{2}}$ is a polynomial (the variable never appears under a root symbol.)