m3-12mn2=40, so n2=(m3-40)/(12m); (m3-12mn2)2=m6-24m4n2+144m2n4=402;
4n3-3m2n=10, so 8n3-6m2n=20; (8n3-6m2n)2=64n6-96m2n4+36m4n2=202;
(m2+4n2)3=m6+3m4(4n2)+3m2(16n4)+64n6=m6+12m4n2+48m2n4+64n6.
Therefore:
m6-24m4n2+144m2n4+64n6-96m2n4+36m4n2=
m6+m4n2(-24+36)+m2n4(144-96)+64n6=
m6+12m4n2+48m2n4+64n6=(m2+4n2)3=402+202=1600+400=2000.
m2+4n2=∛2000.
ALSO:
(m3-12mn2)2+(8n3-6m2n)2=
4n3-3m2n=10=n(4n2-3m2)=√((m3-40)/(12m))((m3-40)/(3m)-3m2), so:
√((m3-40)/(12m))((m3-40)/(3m)-3m2)=10, which is an equation in only one variable.
Without going into details, m=-1.28042 and n=√((m3-40)/(12m))=1.65527 approx.
Therefore m2+4n2=12.5992 approx (∛2000).