<s>[INST] How to factor $4x^2 + 2x + 1$? I want to know how to factor $4x^2 + 2x + 1$? I found the roots using quadratic equation and got $-1 + \sqrt{-3}$ and $-1 - \sqrt{-3}$, so I thought the factors would be $(x - (-1 + \sqrt{-3}))$ and $(x - (-1 - \sqrt{-3}))$
However, according to MIT's course notes, the factors are $(1 - (-1 + \sqrt{-3})x)$ and $(1 - (-1 - \sqrt{-3})x)$  
Course Notes (pg. 30) 
Are these two expressions equivalent, or am I simply not factoring correctly? Thanks.
 [/INST] Given $$\displaystyle 4x^2+2x+1 = \underbrace{(2x)^2+1^2+\left(\frac{1}{2}\right)^2}+1-\underbrace{\left(\frac{1}{2}\right)^2}$$
So we get $$\displaystyle \left(2x+\frac{1}{2}\right)^2-\left(\frac{\sqrt{3}i}{2}\right)^2 = \left(2x+\frac{1}{2}+\frac{\sqrt{3}i}{2}\right)\cdot \left(2x+\frac{1}{2}-\frac{\sqrt{3}i}{2}\right)$$
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