system

convert first into $b=14-a$

then insert into second:

$a^3+(14-a)^3=812+a^2+(14-a)^2$

the end result is

$a=\dfrac{35\pm2\sqrt{35}}{5}$

insert a into first equation:

$b=\dfrac{35\pm2\sqrt{35}}{5}$

so $\boxed{a,b=\dfrac{35\pm2\sqrt{35}}{5}}$