Question Check if the polynomial $$ 2X^{34}Y^{34}+(3X^5Y^3+24X^3Y+9X^2)Y^2+3\in\mathbb{R}[X,Y] $$ is irreducible.
Problem Normally when I asked a question here I always had some clue to do something to the question. But this time I really don't have even a tiny idea how to check this. I tried to consider $\mathbb{R}[X,Y]=(\mathbb{R}[X])[Y]$ or the other way round but could not find a solution. I also tried to add in terms to factorise this, because I have noticed that all monomials in this polynomial have the form $X^kY^k$ for some $k$ except $3$.
Could someone help me with this please?
It's a degree $34(>2)$ polynomial in $Z=XY$ so it's reducible in $\mathbb{R}[XY]$ and therefore reducible in $\mathbb{R}[X,Y]$