The equation $x^2+1 = yzt$ has a parametric solution as follows.

We factorize $x^2+1 = yzt$ in $\mathbb Z[i].$

Let $(y,z,t)=(a+bi,c+di,e+fi)$ then

$x+i = xyz = (acf+ead+ebc-bdf)i+ace-fad-fbc-bde.$

Hence we get a parametric solution $x = acf+ead+ebc-bdf$ if $ace-fad-fbc-bde =1.$

The equation $x^2+1 = yzt$ has a parametric solution as follows.

We factorize $x^2+1 = yzt$ in $\mathbb Z[i].$

Let $(Y,Z,T)=(a+bi,c+di,e+fi)$ then

$x+i = YZT = (acf+ead+ebc-bdf)i+ace-fad-fbc-bde.$

Hence we get a parametric solution $x = acf+ead+ebc-bdf$ if $ace-fad-fbc-bde =1.$