Why in the world would you first **multiply** and then factor? You start with \(\displaystyle -(x- 2)(x^2+ 2x+ 5)\) so obviously, **one** factor is x- 2. In order to factor \(\displaystyle x^2+ 2x+ 5\) into factors with **integer** coefficients, I would note that 5 factors only as 1(5) but neither 5-1 nor 5+ 1 is equal to 2 so we **can't** factor with integer coefficients. Not every quadratic can be factored with integer (or even real) coefficients.

So instead, **complete** the square! \(\displaystyle x^2+ 2x+ 1= (x+ 1)^2\) so that \(\displaystyle x^2+ 2x+ 5= x^2+ 2x+ 1+ 4= (x+1)^2+ 4= (x+1)^2- (-4)= (x+ 1)^2- (2i)^2\). Now use the identity \(\displaystyle a^2- b^2= (a- b)(a+ b)\).