First of all ,
For p to lie between roots of equation the discriminant of given quadratic equation > 0 as roots must exist and they must be distinct..
So ,
[2(p-3)]2 - 4.9 > 0
==> 4(p-3)2 - 4 . 9 > 0
==> (p-3)2 - 9 > 0
==> p > 6 or p < -6
Also , if we require specifically that 6 lies in between the roots of the equation , and here a = 1 in the quadratic equation ax2 + bx + c = 0 . So as we know in such case the graph is an upward parabola..So
For 6 to lie between the roots , f(6) < 0 is necessary condition ..So substituting x = 6 in the given quadratic polynomial we have :
62 + 2.(p-3).6 + 9 < 0
==> 36 + 12 p - 36 + 9 < 0
==> 12 p + 9 < 0
==> p < -3 / 4
So according to discriminant value requirement p > 6 or p < -6 and here p < -3 / 4 ..
So taking common case we have p < -6 which will satisfy both the properties..
Hence the range of p is : ( -infinity , 6 )