Solution - Simplification or other simple results

Step  1  :

Trying to factor by splitting the middle term

 1.1     Factoring  -x²+2x+3 

The first term is,  -x²  its coefficient is  -1 .
The middle term is,  +2x  its coefficient is  2 .
The last term, "the constant", is  +3 

Step-1 : Multiply the coefficient of the first term by the constant   -1 • 3 = -3 

Step-2 : Find two factors of  -3  whose sum equals the coefficient of the middle term, which is   2 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -1  and  3 
                     -x² - 1x + 3x + 3

Step-4 : Add up the first 2 terms, pulling out like factors :
                    -x • (x+1)
              Add up the last 2 terms, pulling out common factors :
                    3 • (x+1)
Step-5 : Add up the four terms of step 4 :
                    (-x+3)  •  (x+1)
             Which is the desired factorization

Final result :

  (x + 1) • (3 - x)