Step 1: Factor the constant term. The factors of 19 are -1, 1, -19, 19. Substitute each of those into the equation (x^3 - 7x^2 + 11x + 19=0). If the number makes the equation true then it is root and x - the number is a factor.
Substitution the numbers back into the equation we find -1 makes the equation true, so x+1 is a factor.
Use synthetic division to find what remains.
-1 | 1 -7 11 19 ---> carry down the first 1...-1*1 = -1 and -7-1=-8....-8*-1 = 8 and 11+8 =19....-1*19=-19 and 19-19=0. The remaining polynomial is x^2 -8x +19.
Factor x^2 -8x +19 using the quadratic formula to find it has only complex roots. In other words, it is already in simplest form.
The answer is then (x+1)(x^2 -8x +19)
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