Pls guve me sol. Factors?

Dear Student,

Please find below the solution to the asked query:

We have quadratic equation :  x² + 2 x  +  3 

We know quadratic formula :  x  = $\frac{- \mathrm{b} \pm \sqrt{{\mathrm{b}}^2 - 4\mathrm{ac}}}{2 \mathrm{a}}$ , Here a = 1 , b  = 2 and c  = 3 

So,

x  = $\frac{- 2 \pm \sqrt{2^2 - 4\times 1 \times 3}}{2 \times 1} = \frac{- 2 \pm \sqrt{4 - 12}}{2}= \frac{- 2 \pm \sqrt{- 8}}{2}= \frac{- 2 \pm 2 i\sqrt{2}}{2}= \mathbf{-}\mathbf{ }\mathbf{1}\mathbf{ }\mathbf{\pm }\mathbf{ }\mathit{i}\sqrt{\mathbf{2}}$
Then,

Factors  =  ( x - ( - 1 - i$\sqrt{2}$ ) = ( x + 1 + i$\sqrt{2}$ )

And

( x - ( - 1 + i$\sqrt{2}$ ) = ( x + 1 - i$\sqrt{2}$ )

Therefore,

x² + 2 x  +  3   = ( x + 1 + i$\sqrt{2}$ ) ( x + 1 - i$\sqrt{2}$ )                                              ( Ans )

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