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Find x value in quartic equation

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To solve the quartic equation, one of the approaches is to seek squares on both sides. For example, to find x, in ax^4+bx^2+cx=d. One way is to convert the equation to the following format: (square root of a x^2+e)^2=(gx+h)^2. However, when a, e, g and b becomes irrational number, it is difficult to apply this strategy. Here, I provide a common solution for this type of equation with a specific example.

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The solution provides a theory how to find x value in quartic equation.

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solve x^4 + 10x = 5x^2 + 5.
First, x^4-5^2+10x-5=0. (I)
we factor the left side: (x^2+bx+c)(x^2+dx+e)=0. (II)
x^4+(d+b)x^3+(c+e)x^2+(be+cd)x+ce=0.
when we compare equation (I) and ...

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