linear algebra: a quadratic factor in two variables can be factored if the determinant is $ 0 $

(All real numbers)
Show that
$$ ax ^ 2 + 2hxy + by ^ 2 + 2gx + 2gy + c = 0 $$
can be factored as
$$ (a_1x + b_1y + c_1) (a_2x + b_2y + c_2) = 0 $$
iff
$$ begin {vmatrix} a & h & g \ h & b & f \ g & f & c end {vmatrix} = 0 $$


I have seen people who use this as a fact without any proof. For example, in this video (sorry, not in English) he declares it as a fact without any explanation, which is really frustrating. I wonder if there is a way to make sense of this using linear algebra or calculation. We greatly appreciate any help. Thank you!