10. The common difference of an arithmetic sequence with real terms is 4, and the square of the first term plus the sum of the remaining terms does not exceed 100. Such a sequence can have at most $\qquad$ terms.
Let the sequence $\left\{a_{n}\right\}$ have the first term $a_{1}$, and the common difference be $4 \Rightarrow S_{n}=n a_{1}+2 n(n-1)$
$$
\begin{array}{l}
\Rightarrow a_{1}^{2}+S_{n}-a_{1} \leqslant 100 \Rightarrow a_{1}^{2}+(n-1) a_{1}+2 n^{2}-2 n-100 \leqslant 0 \\
\Rightarrow\left(a_{1}+\frac{n-1}{2}\right)^{2}+\frac{7}{4} n^{2}-\frac{3}{2} n-\frac{401}{4} \leqslant 0 \Rightarrow 7 n^{2}-6 n-401 \leqslant 0 \\
\Rightarrow n \leqslant\left[\frac{3+\sqrt{9+7 \cdot 401}}{7}\right]=8 . \text { Therefore, such a sequence can have at most } 8 \text { terms. }
\end{array}
$$