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?

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.
