Is it possible to evaluate $232^2-62^2\times14$ by factoring or using identities or...? The expression $232^2-62^2\times14$ can be calculated directly ($53824-3844
\times14=8$). But is it possible to evaluate it for example by factoring or using identities?
Here is what I have tried,
$$(58\times4)^2-62^2\times14=58^2\times16-62^2\times14=(60-2)^2(15+1)-(60+2)^2(15-1)$$
Or
$$58^2\times16-62^2\times14=29^2\times64-31^2\times56=(30-1)^2\times8^2-(30+1)^2(8\times7)$$
But I can't see an elegant way to get $8$ from either of the calculatins.

this is pretty similar to the answer by lhf.   By solving a Pell equation $u^2 - 14 v^2 = 1$   we find formulas:
$$    x^2 - 14 y^2 = (15x + 56y)^2  - 14 (4x+15y)^2  \; ; \;\; $$
or the reverse direction
$$    x^2 - 14 y^2 = (15x - 56y)^2  - 14 (-4x+15y)^2  \; ; \;\; $$
If $(x,y) = (232, 62), $  the reverse formula says
$$    232^2   - 14 \cdot 62^2  = 8^2 - 14 \cdot 2^2  $$
and this is $64 - 56 = 8$
