(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!