If equations ax² + bx + c = 0, (a, b, c ∈ R, a ≠ 0) and 2x² + 3x + 4 = 0 have a common root, then a:b:c equals:

Consider the equation : 2x² + 3x + 4 = 0
Let α be the common root.
Then 2α² + 3α + 4 = 0
Also, aα² + bα + c = 0
From the first equation, 2α² = -3α - 4
α² = (-3α - 4)/2
Substituting this into the second equation, we get:
a((-3α - 4)/2) + bα + c = 0
-3aα/2 - 4a/2 + bα + c = 0
(-3a/2 + b)α + (-2a + c) = 0
Since α is a root of both equations, it must satisfy both equations. For this to be true for any α, the coefficients of α must be equal to 0, and the constant terms must be equal to 0.
Therefore, we have the following system of equations:
-3a/2 + b = 0
-2a + c = 0
From the first equation: b = 3a/2
From the second equation: c = 2a
Thus, a:b:c = a : 3a/2 : 2a = 2 : 3 : 4