  In this note, we use the new bottom up method based on soft theorems to
construct the expansion of single-trace Yang-Mills-scalar amplitudes
recursively. The resulted expansion manifests the gauge invariance for any
polarization carried by external gluons, as well as the permutation symmetry
among external gluons. Our result is equivalent to that found by Clifford
Cheung and James Mangan via the so called covariant color-kinematic duality
approach.
