How to solve equations to the fourth power?

Is it possible to manually retrieve the value of $y$ from the following equation $$\color{blue}{153y^2-y^4=1296}$$

WolframAlpha has four solutions for $y$: $-12, -3, 3, 12$. How has it solved?

What I've achieved to until now is the following: $$y^2(153-y^2)=1296$$ And... I'm stuck.

One method is to set $x=y^2$ and rearrange this as a quadratic equation $$x^2-153x+1296=0$$

Here's the solution:

$$y^4-153y^2 +1296 = 0$$

$$ y^4 -144y^2-9y^2+1296 = 0$$

$$ y^2(y^2-144) -9(y^2-144) = 0$$

$$ (y^2-9)(y^2-144)=0$$ $$ (y^2-3^2)(y^2-12^2)=0$$

note that $a^2 - b^2 = (a-b)(a+b)$

Can you see how now? I trust you can finish the rest