<s>[INST] How to solve this problem on complex numbers? Let $ Z $ be a complex number with nonzero imaginary part such that
$$ (2Z + 1)(3Z + 1)(5Z + 1)(30 Z + 1) = 10 $$
Then compute  $$ \frac{\text {sum  of  all values  of  Z}}{\text {product  of  all  values  of  Z} } $$
 [/INST] Hint: $1/2+1/30=1/3+1/5 = 2 \cdot 4/15\,$, which suggests the substitution $z = x - 4/15\,$, which gives a biquadratic in $x\,$:
$$
2\cdot3\cdot5\cdot30\left(z+\frac{1}{2}\right)\left(z+\frac{1}{3}\right)\left(z+\frac{1}{5}\right)\left(z+\frac{1}{30}\right) \\ = 900\left(x+\frac{7}{30}\right)\left(x+\frac{1}{15}\right)\left(x-\frac{1}{15}\right)\left(x-\frac{7}{30}\right) \\ = 900\left(x^2-\frac{7^2}{30^2}\right)\left(x^2-\frac{1}{15^2}\right) \\ = \frac{1}{225}(900x^2-49)(225x^2-1)
$$
Then $225(\text{LHS} - 10)$ factors into two qadratics, one of which has the complex roots being sought:
$$
(900x^2-49)(225x^2-1)-2250=(900 x^2 + 71) (225 x^2 - 31)
$$
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