Can we fully factorise $x^4+4y^4$?

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Let’s start from the left-hand side and multiply out the brackets. We find \[\begin{align*} (x^2+2y^2)^2-4x^2y^2 &=x^4+4x^2y^2+4y^4-4x^2y^2\\ &=x^4+4y^4, \end{align*}\]

which is what we wanted.

We know that \[ x^4+4y^4=(x^2+2y^2)^2-4x^2y^2. \] Notice that the right hand side is of the form \(a^2-b^2\), where \(a=x^2+2y^2\) and \(b=2xy\).

We know that \(a^2-b^2=(a+b)(a-b)\), and so \[ (x^2+2y^2)^2-(2xy)^2=(x^2+2y^2+2xy)(x^2+2y^2-2xy). \]

Can we factorise the terms in the brackets any further? We can think of the first bracket as \[ x^2+2yx+2y^2, \] which is a quadratic in \(x\) with coefficients that depend on \(y\). If this factorised into \((x+ay)(x+by)\) for some \(a\) and \(b\), then there would be one or two real solutions to \(x^2+2yx+2y^2=0\) for each real value of \(y\). So we can check the discriminant,
\[ (2y)^2-4(2y^2)=4y^2-8y^2=-4y^2. \] But this is less than zero whenever \(y\) is non-zero. Thus \(x^2+2y^2+2xy=0\) doesn’t have any real solutions if \(y\ne0\), and so we can’t factorise the first bracket any further.

For the second bracket, the discriminant is also \[ (-2y)^2-4(2y^2)=-4y^2, \] and so we can’t factorise the second bracket either. Therefore we’ve fully factorised \(x^4+4y^4\).