All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(i)
$x^2 -6x + 34 = 0$

$x = \frac{6 \pm \sqrt{36-136}}{2}$

$x = 3 \pm 5i$

(ii)
Since the coefficients are all real, another root of the equation is $-2-i$

$[x-(-2+i)][x-(-2-i)]$

$= x^2 + 4x +5$

By comparing coefficients,

$x^4 + 4x^3 + x^2 + ax + b = (x^2+4x+5)(x^2-4)$

$\therefore, a=-16, b = -20$

The other three roots are $-2 - i, 2, -2$