x^4+x^2-6=(x^2+3)(x^2-2), and 4x^2-28=4(x^2-7).
4[(x^2-7)/(x^4+x^2-6)]=4[(Ax+B)/(x^2+3)+(Cx+D)/(x^2-2)]
4[(Ax+B)(x^2-2)+(Cx+D)(x^2+3)]=4[(Ax^3-2Ax+Bx^2-2B)+(Cx^3+3Cx+Dx^2+3D)]
Equating coefficients to match 4(x^2-7):
(1) A+C=0 (x^3)
(2) B+D=1 (x^2)
(3) -2A+3C=0 (x)
(4) -2B+3D=-7 (constant)
2(1)+(3): 5C=0, so A=C=0 (as expected because there are no odd powers of x anywhere)
2(2)+(4): 5D=-5, D=-1, B=2
4(x^2-7)=4(2/(x^2+3)-1/(x^2-2)).