$P(x)$ is a formula, but not a statement.
Formulas can have free variables, and indeed $P(x)$ has a free (unquantified) variable $x$. Because it is not quantified, we cannot assign a truth-value to it, and so it is not a statement.
Indeed, note that I can quantifiy the $x$ either existentially or universally (or, if I had different quantifiers, differently yet). But, $\exists x \ P(x)$ is clearly a different statement than $\forall x \ P(x)$: I can easily come up with a domain and interpretation of $P(x)$ where $\exists x \ P(x)$ is true but $\forall x \ P(x)$ is false. So, I can indeed not assign a truth-value just to $P(x)$, so it is not a statement.
$P(x)$ is a formula though: it does follow the syntactical rules (the 'grammar' if you want) for expressions in FOL. I don;t have the text, but presumably the author lays out exactly those rules in defining formulas, and you can check for yourself that $P(x)$ does follow that definition. We sometimes therefore call formulas 'well-formed formulas' or wff's for short.
Something like $)xP)((,)$ is clearly not a formula ... that's just a string of symbols, but otherwise gibberish