So, I suppose we have a matrix $A\in\Bbb R^{N\times M}$, and we want to solve $Ax=b$ Then, indeed, if $x_p$ is any solution (named 'particular'), then $A(x-x_p)=0$ is remained to solve, hence in this case, all the solutions are of the form $x_p+x_n$ as you write, where $x_n$ is in the nullspace of $A$.
On the other hand, according to the textbook, every vector $x$ can be written as $x_r+x_n$ where $x_r$ is in the rowspace of $A$ and $x_n$ is in the nullspace. (For this, consider the rowspace and project $x$ orthogonally to it.)
The common point in the above ones is the 'nullspace', and thus we can state not more than
An $x$ is a solution iff its row space component, $x_r$ is a solution.