What Are the Real Solutions for This Equation?

[anemone]

Solve for real solution(s) for $$x^2− x + 1 = (x^2+ x + 1)(x^2+ 2x + 4)$$.

 

[MarkFL]

Solve for real solution(s) for $$x^2− x + 1 = (x^2+ x + 1)(x^2+ 2x + 4)$$.

My solution:

Spoiler

If we expand and collect like terms, we have:

$$x^4+3x^3+6x^2+7x+3=0$$

Let's assume the LHS factors in the form:

$$x^4+3x^3+6x^2+7x+3=(x^2+ax+1)(x^2+bx+3)=x^4+(a+b)x^3+(ab+4)x^2+(3a+b)x+3$$

Equating coefficients selected to result in a linear 2X2 system, we have:

$$a+b=3$$

$$3a+b=7$$

From this we find:

$$(a,b)=(2,1)$$

Hence:

$$x^4+3x^3+6x^2+7x+3=(x^2+2x+1)(x^2+x+3)=(x+1)^2(x^2+x+3)=0$$

The discriminant of the second factor is negative, thus the only real solution is:

$$x=-1$$