t=(x+y)(y+z)(z+x)

Solution for t=(x+y)(y+z)(z+x) equation:

Simplifying
t = (x + y)(y + z)(z + x)

Reorder the terms:
t = (x + y)(y + z)(x + z)

Multiply (x + y) * (y + z)
t = (x(y + z) + y(y + z))(x + z)
t = ((y * x + z * x) + y(y + z))(x + z)
t = ((xy + xz) + y(y + z))(x + z)
t = (xy + xz + (y * y + z * y))(x + z)

Reorder the terms:
t = (xy + xz + (yz + y2))(x + z)
t = (xy + xz + (yz + y2))(x + z)
t = (xy + xz + yz + y2)(x + z)

Multiply (xy + xz + yz + y2) * (x + z)
t = (xy(x + z) + xz(x + z) + yz(x + z) + y2(x + z))
t = ((x * xy + z * xy) + xz(x + z) + yz(x + z) + y2(x + z))

Reorder the terms:
t = ((xyz + x2y) + xz(x + z) + yz(x + z) + y2(x + z))
t = ((xyz + x2y) + xz(x + z) + yz(x + z) + y2(x + z))
t = (xyz + x2y + (x * xz + z * xz) + yz(x + z) + y2(x + z))

Reorder the terms:
t = (xyz + x2y + (xz2 + x2z) + yz(x + z) + y2(x + z))
t = (xyz + x2y + (xz2 + x2z) + yz(x + z) + y2(x + z))
t = (xyz + x2y + xz2 + x2z + (x * yz + z * yz) + y2(x + z))
t = (xyz + x2y + xz2 + x2z + (xyz + yz2) + y2(x + z))
t = (xyz + x2y + xz2 + x2z + xyz + yz2 + (x * y2 + z * y2))
t = (xyz + x2y + xz2 + x2z + xyz + yz2 + (xy2 + y2z))

Reorder the terms:
t = (xyz + xyz + xy2 + xz2 + x2y + x2z + yz2 + y2z)

Combine like terms: xyz + xyz = 2xyz
t = (2xyz + xy2 + xz2 + x2y + x2z + yz2 + y2z)

Solving
t = 2xyz + xy2 + xz2 + x2y + x2z + yz2 + y2z

Solving for variable 't'.

Move all terms containing t to the left, all other terms to the right.

Simplifying
t = 2xyz + xy2 + xz2 + x2y + x2z + yz2 + y2z