Its actually simple. Observe that Span(3x, 2 + 3/2x^2,4x+x^3) has a dimension of 3. Observe that the rank is 3.
maps to . Thus matrix of Transformation is
Its easy to check that the rank of the above matrix is 3.
Hence dimension(image) = rank = 3.
Thus by rank-nullity theorem nullity = 1 and thus the kernel has a dimension 1. This also implies that the Transform is not one-one.