Ta có: x-2 là ước của \(2x^2+19\)
⇔\(2x^2+19⋮x-2\)
⇔\(2x^2-4x+4x+19⋮x-2\)
mà \(2x^2-4x=2x\left(x-2\right)⋮x-2\)
nên \(4x+19⋮x-2\)
hay \(4x-8+27⋮x-2\)
mà \(4x-8=4\left(x-2\right)⋮x-2\)
nên \(27⋮x-2\)
⇔\(x-2\inƯ\left(27\right)\)
\(\Leftrightarrow x-2\in\left\{1;-1;3;-3;9;-9;27;-27\right\}\)
\(\Leftrightarrow x\in\left\{3;1;5;-1;11;-7;29;-25\right\}\)(tm)
Vậy: \(x\in\left\{3;1;5;-1;11;-7;29;-25\right\}\)