To find out A, B and C, we have to use the three equalities.
To find A, expand the first equality. Substitute using the other 2 equalities when necessary.
To find B, expand (x^2+y^2+z^2)^2. Rearrange the expanded equation and you can get B eventually.
From the question, we know that x^4+y^4+z^4 = C. So, we can rewrite the equation (x^2+y^2+z^2)^2 = C + B[(xy)^2+(yz)^2+(zx)^2)]. And the tricky part now is to find [(xy)^2+(yz)^2+(zx)^2)] which is by expanding the equation (xy+yz+xz)^2. Substitute when necessary and you will get the answer C.
Sorry that my explaination might be difficult to understand. Do ask if you have more questions.