Its the white space after class=
. Try this:
private static void ContainsTest()
{
string input = "<a class=";
string test = "<a class=\"doctor\" href=\"/public-register/doctor-details.aspx?view=1&id=%2082358\">Aal Ali, Saleh Saif Salem A S Fares </a>(#82358)";
Console.WriteLine("test Contains input: "+test.Contains(input).ToString());
}