Factoring quadratics with certain property I am looking into which quadratics factor with the following property:
$x^2±ax±b$ factors e.g. all the following factor for $a=5, b=6$
$x^2+5x+6=(x+2)(x+3)$
$x^2-5x+6=(x-2)(x-3)$
$x^2+5x-6=(x+6)(x-1)$
$x^2-5x-6=(x-6)(x+1)$
I have found $x^2+10x+24, x^2+20x+96, ..., x^2+5 \cdot 2^{n-1}x+6\cdot 4^{n-1}$ all have this property. I am looking for others or is this the only family of MONIC quadratics that have this property?

You're looking for $a^2\pm 4b$ to be perfect squares, say$$a^2\pm 4b=(c\pm d)^2=c^2+d^2\pm 2cd\iff a^2=c^2+d^2,\,b=cd/2.$$So this reduces to Pythagorean triples viz.$$c=2klm,\,d=(k^2-l^2)m,\,a=(k^2+l^2)m,\,b=klm^2(k^2-l^2).$$In your example $k=2,\,l=m=1$.
