By a {\it diagonal minus tail } form (of even degree) we understand a real homogeneous polynomial $F(x_1,\ldots,x_n)=F(x)=D(x)-T(x),$ where the {\it diagonal} part $D(x)$ is a sum of terms of the form $b_i x_i^{2d}$ with all $b_i\geq 0$ and the {\it tail} $T(x)$ a sum of terms $a_{i_1i_2\cdots i_n}x_1^{i_1}\cdots x_n^{i_n}$ with $a_{i_1i_2\cdots i_n} > 0$ and at least two $i_\nu\geq 1.$ We show that an arbitrary change of the signs of the tail terms of a positive semidefinite diagonal minus tail form will result in a sum of squares of polynomials. We also give an easily tested sufficient condition for a polynomial to be a sum of squares of polynomials (sos) and show that the class of polynomials passing this test is wider than the class passing Lasserre's recent conditions. Another sufficient condition for a polynomial to be sos, like Lasserre's piecewise linear in its coefficients, is also given. This work was completed in 2011. We also will report on refinements found a year later by Ghasemi and Marshall.