Filter datagridview with research key and only show the last five rows .

Firstly never ever use string concatenation to create your SQL statements like this. Research Parameters and use them.

To get the last 5 rows try this query

SELECT * FROM (
    SELECT * FROM yourTtable ORDER BY (id) DESC LIMIT 5
    WHERE Convert(researchkey,'System.String') like @filter
) subq
ORDER BY (id) ASC

[Edit] - This solution only works when populating the DataGridView and not when filtering. I am attempting to find the correct solution and will revisit this solution as soon as possible.

[Edit2]. I've played around with this for some time and can only limit the display to the first 5 rows which fit the filter - by implementing paging on the datagridview
(These articles give ideas how to do that:
DataGridView With Paging (UserControl)[^] or
A Simple Way for Paging in DataGridView in WinForm Applications[^])

You could display only the last 5 by navigating to the last row in the filtered grid:

DataGridView1.FirstDisplayedScrollingRowIndex = DataGridView1.RowCount - 1
DataGridView1.Rows(Me.DataGridView1.RowCount - 1).Selected = True

Personally I would not do the filtering on the DataGridView but just populate it with pre-filtered data - in which case my original SQL statement would apply.

E.g. (untested)

private void button1_Click(object sender, EventArgs e)
  {
      var constring = ""; //put your connection string in here
      var sql =
          "SELECT * FROM (SELECT * FROM yourTtable ORDER BY (id) DESC LIMIT 5 WHERE Convert(researchkey,'System.String') like @filter) subq ORDER BY (id) ASC";

      using (var connection = new SqlConnection(constring))
      {
          connection.Open();

          using (var myAdapter = new SqlDataAdapter(sql, connection))
          {
              myAdapter.SelectCommand.Parameters.AddWithValue("@filter", textBox1.Text);
              var ds = new DataSet();
              myAdapter.Fill(ds);
              bs.DataSource = ds.Tables[0];
          }

          connection.Close();
      }
  }

  private void button1_Click(object sender, EventArgs e)
        {
            String constring = "";
            MySqlConnection connection = new MySqlConnection(constring);
            connection.Open();
            BindingSource bs = new BindingSource();

            bs.DataSource = dataGridView1.DataSource;
            bs.Filter = "SELECT * FROM (SELECT * FROM database.table ORDER BY (id) DESC LIMIT 5 WHERE Convert(researchKey,'System.String') like '" + textbox.Text + "') subq ORDER BY (id) ASC";

       
            dataGridView1.DataSource = bs;

        }