My lectures notes are showing that:
$$c^{T}x + v^{T}(Ax-b) - \lambda^{T}x = -b^{T}v + (c+A^{T}v - \lambda)^{T}x$$
However, I cannot see how we we need to use the property that the transpose of the sum is the sum of the transpose. My way of thinking about that is:
$$c^{T}x + v^{T}(Ax-b) - \lambda^{T}x = (c^{T}+v^{T}A- \lambda^{T})x - v^{T}b$$
Wouldn't that be correct too?