GAIC Math 2024 Problem 36

Let $a_i, b_i \ (i=1,2,\ldots,n)$ be rational numbers such that for any real number $x$, we have $x^2 + x + 4 = \sum_{i=1}^{n} (a_{i}x + b_{i})^{2}$. Then the minimum possible value of $n$ is \underline{\hspace{2cm}}.