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First, apply the Rational Root Theorem to test possible rational roots, which are factors of the constant term \(-4\) over factors of the leading coefficient \(2\): \( \pm 1, \pm 2, \pm 4, \pm rac{1}{2}, \pm rac{1}{4} \).
m(1) = 2(1)^3 - 9(1)^2 + 12(1) - 4 = 2 - 9 + 12 - 4 = 1
m(2) = 2(2)^3 - 9(2)^2 + 12(2) - 4 = 16 - 36 + 24 - 4 = 0

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