<s>[INST] Find all values of the real parameter $a$ for which the equation $4x^4+(8+4a)x^3+(a^2+8a+4)x^2+(a^3+8)x+a^2=0$ has only real roots 
Find all values of the real parameter a for which the equation $$4x^4+(8+4a)x^3+(a^2+8a+4)x^2+(a^3+8)x+a^2=0$$ has only real roots.

Obviously as soon as you factor this equation to $$(4x^2+8x+a^2)(x^2+ax+1)=0$$ then you have finished.
However, how am I supposed to think of factoring the equation in such a manor? That you are supposed to factor it, is obvious, but how can you find out which are its factors, aside from blind luck?
 [/INST] Rewrite it as a polynomial in $a$:
$$ a^3 x + a^2 (x^2 + 1) + a ( 4x^3 + 8x^2 ) + (4x^4 + 8x^3 + 4x^2 + 8x) = 0.$$
The factorization almost immediately jumps out by observation:
$$ (a^2 + 4x^2 + 8x ) ( ax + x^2 + 1) = 0 $$

The idea of changing the variable is a common trick. It can be helpful when factoring (e.g. applying the Reminder Factor Theorem), or finding real roots (e.g. apply the quadratic discriminant to another variable).
As an example, try to factorize $ a^4 + b^4 + c^4 - 2a^2b^2 - 2b^2c^2 - 2c^2a^2$.
There are numerous approaches that one can use, so in the spirit of this question, consider it as a quadratic in $a^2$ first.
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