Transform the equation by completing the square:
4x^2 + y^2 - 8x + 6y -3 =0
4x^2 - 8x + y^2 + 6y - 3 = 0
4(x^2 - 2x) + (y^2 + 6y) = 3
Now add constants to make perfect squares in the parentheses. Add the same constants on the other side of the equation.
4(x^2 - 2x + 1) + (y^2 + 6y + 9) = 3 + 4*1 + 9
4(x-1)^2 + (y+3)^2 = 16
(x-1)^2/2^2 + (y+3)^2/4^2 = 1
Looks like an ellipse with center at (1,-3)
major axis along the line x=1
semi-axis lengths: 2 and 4