Writing a polynomial as a product of two other polynomials?

Question

Ok, I have no idea how to do this, or really what even its asking. The original question wasn't in English, so I apologize if the vernacular is off.

We have a polynomial $P(X) = X^4 - 3X^2 - 4$ (of $\mathbb R[X]$). I need to write $P(X)$ as a product of two polynomials of degree 1 and a polynomial of degree 2 with a negative discriminant.

It also gives me a hint: To search for the roots, we can use $Z = X^2$ for the time being.

I wish I could say more, and give you the way that I attempted solved this, but I legitimately have no idea what to do here. Any help is appreciated, thanks.

Answer 1

Let's write $X^2=y$, so that our polynomial is $y^2-3y-4$. We are used to factoring quadratics, so we see that $y^2-3y-4=(y-4)(y+1)=(X^2-4)(X^2+1)$.

Now $(x^2-4)=(X-2)(X+2)$ and $(X^2+1)$ has negative discriminant.

The only tricks here are to recognize that you can factor the quartic as if it were a quadratic, and then also recognize the difference of squares $X^2-2^2=(X-2)(X+2)$.

Answer 2

You have a quadratic-in-drag. Notice that $$x^4 - 3x^2 - 4 = (x^2 - 4)(x^2 + 1). $$

Answer 3

Your attempt to substitute $X^2 = Z$ is a good start. Now try to solve the equation $$X^4 - 3 X^2 -4 = 0$$ $$Z^2 - 3Z - 4 =0$$ $$(Z-4)(Z+1) = 0$$

And then fill in $X^2 = Z$ and factor everything further. Do you think you can take if from here?