What is the domain and range of f(x)=2x^2+6x+2

As is the case for any polynomial, the domain of this one is (-infinity,  +infinity).

To find the range, we need to determine the minimum value that f(x) can have.  The coefficients here are a=2, b=6 and c = 2,

The x-coordinate of the vertex is  x = -b/(2a), which here is x = -6/4 = -3/2.

Evaluate the function at x = 3/2 to find the y-coordinate of the vertex, which is also the smallest value the function can take on.  That happens to be y = -5/2, so the range is [-5/2, infinity).