solve by using the quadratic formula prove your answer is correct by checking

plug and chug:

x = (-3±√(3^2-4*2*4))/(2*2)
= (-3±√(9-32))/4
= (-3±i√23)/4

check:

2(-3/4+√23/4 i)^2 + 3(-3/4+√23/4 i) + 4
2(9/16 - 3√23/8 i - 23/16) - 9/4 + 3√23/4 i + 4
= 9/8 - 3√23/4 i - 23/8 - 9/4 + 3√23/4 i + 4
= 0

similarly for -3/4-√23/4 i