Start with
[tex](a+b+c)^3=a^3+b^3+c^3+3(a^2b+a^2c+ab^2+ac^2+b^2c+bc^2)+6abc[/tex]
Given that [tex]a+b+c=5[/tex], we know [tex](a+b+c)^3=125[/tex].
Also, [tex]ab+bc+ca=10[/tex], so
[tex]a^2b+a^2c+ab^2+ac^2+b^2c+bc^2=a(ab+ac)+b(ab+bc)+c(ac+bc)[/tex]
[tex]=a(10-bc)+b(10-ca)+c(10-ab)[/tex]
[tex]=10(a+b+c)-3abc[/tex]
[tex]=50-3abc[/tex]
So we have
[tex]125=a^3+b^3+c^3+3(50-3abc)+6abc[/tex]
[tex]125=a^3+b^3+c^3+150-9abc+6abc[/tex]
[tex]\implies -25=a^3+b^3+c^3-3abc[/tex]
QED
