<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] Observing and categorizing the coefficients can be effective here.
Rewrite the equation into
$$(4x^4+8x^3+a^2x^2)+(4ax^3+8ax^2+a^3x)+(4x^2+8x+a^2)=0\\\Longrightarrow x^2(4x^2+8x+a^2)+ax(4x^2+8x+a^2)+(4x^2+8x+a^2)=0\\\Longrightarrow (x^2+ax+1)(4x^2+8x+a^2)=0$$
Or in this way:
$$(4x^4+4ax^3+4x^2)+(8x^3+8ax^2+8x)+(a^2x^2+a^3x+a^2)=0\\\Longrightarrow 4x^2(x^2+ax+1)+8x(x^2+ax+1)+a^2(x^2+ax+1)=0\\\Longrightarrow (4x^2+8x+a^2)(x^2+ax+1)=0$$
Does this help? I'm not sure if you are asking for a general way or  just for this one problem.
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