A conic in the plane $\mathbb{R}^2$ is the zero set of a quadratic polynomial in two variables:
$$ a_1 x^2 + a_2 xy + a_3 y^2 + a_4 x + a_5 y + a_6 .$$
Geometrically, a conic can be either a circle, an ellipse, a hyperbola, a parabola or a union of two lines. The last case is called a degenerate conic.
Steiner's conic problem asks the question of how many conics are tangent to five given conics. This webpage gives an answer to a slightly different question:
Which conics are tangent to your five conics?
Plug in the coefficients of your five personal conics, and get the answers in a second (your conics must be sufficiently generic in the sense that the 3264 complex conics tangent to your five conics are all isolated; in particular, degenerate conics are not allowed).
More information on how this works can be found in the article 3264 conics in a Second.