2022/030) Factor $x^4+3 x^2 y^2+2 y^4+4 x^2+5 y^2+3$

we have

 $x^4+3 x^2 y^2+2 y^4+4 x^2+5 y^2+3$

$= x^4+3 x^2 y^2+4 x^2 + 2 y^4 +5 y^2+3$ putting them in descending power of x

$=x^4+ x^2(3 y^2 +4) +  (2y^2 + 1)(y^2 + 3)$ factoring the expression involving y only

$=x^4+ x^2((2y^2 + 1) + (y^2 +3) +  (2y^2 + 1)(y^2 + 3)$ split the middle term as $3y^2 + 4 = 2y^2 + 1 + y^3 + 3$

$= x^2(x^2 + 2y^2 + 1) + (y^2 + 3)(x^2 + 2y^2  +1 ) = (x^2 +2y^2 + 1)(x^2 + y^2 + 3)$