f(x)=x(2x4+7x3+7x2+8x+12) because x is a common factor.
Now we try to identify rational zeroes by factorising 12, which has factors 1, 2, 3, 4, 6, 12.
f(1)≠0, f(-1)≠0, f(2)≠0, f(-2)=0. So x=-2 is a zero and we can divide by this zero using synthetic division:
-2 | 2 7 7 8 1
2 -4 -6 -2 | -12
2 3 1 6 | 0 = 2x3+3x2+x+6.
It happens that -2 is a zero again:
-2 | 2 3 1 6
2 -4 2 | -6
2 -1 3 | 0 = 2x2-x+3.
Now to find the complex zeroes.
2x2-x+3=0,
2(x2-x/2)=-3,
2(x2-x/2+1/16)=-3+1/8=-23/8,
2(x-¼)2=-23/8,
(x-¼)2=-23/16,
x-¼=±i√23/4, x=¼±i√23/4. The complex zeroes are x=¼+i√23/4, x=¼-i√23/4.
The real multiple zero is x=-2 (twice).