\(x^2 + nx + 15 < -89 - 3x^2\)

so i first brought everything to the left side to get \(4x^2 + nx + 104 < 0\)

then i divided by 4 to get\(x^2 + \frac{n}{4}x + 26 < 0\)

for x to have no real solutions the discriminant must be negative

so \(\frac{n^2}{16}-4(26)\) must be negative. the less than sign is useless since smaller right hand values make c greater

the greatest value of n that works is 40

but we also can have negatives, least of which is -40

so -40 to 40 is \(\boxed{81}\) numbers