(x2+4x+3)(x2+12x+35)-9=
x4+12x3+35x2
4x3+48x2+140x
3x2 +36x+105-9=
x4+16x3+86x2+176x+96. Look for rational zeroes.
96 has factors 2, 3, 4, 6, 8, 12, 16, 24, 32, 48. Rational zeroes could be the negative of some of these. They're negative because all coefficients are positive. We need to test these in order. -2 is not a zero, so try -4:
-4 | 1 16 86 176 96
1 -4 -48 -152 | -96
1 12 38 24 | 0, -4 is a zero.
Try -4 again because 4 is a factor of 24:
-4 | 1 12 38 24
1 -4 -32 | -24
1 8 6 | 0, -4 is again a zero leaving x2+8x+6 which can be solved:
x2+8x+6=0,
x2+8x+16-10=0,
(x+4)2=10,
x+4=±√10, x=-4±√10=-0.8377 or -7.1623 approx.
So x=-4, -0.8377 or -7.1623.