Set $P = 10a^2 + 14ab - 21bc - 15ca$.
(1) Factorizing $P$, we obtain
$$P = (\mathbf{A}a + \mathbf{B}b)(\mathbf{C}a - \mathbf{D}c).$$
(2) If $5a = \sqrt{6}$, $14b = \sqrt{2} + \sqrt{3} - \sqrt{6}$ and $15c = \sqrt{12} - \sqrt{8}$, then
$$P = \frac{\mathbf{E}}{\mathbf{E} + \mathbf{F}} \frac{\mathbf{G}}{\mathbf{H}}$$
and hence the greatest integer less than $P$ is $\mathbf{I}$.