Question

Factor completely −7x2 + 14x − 21.

Answer 1

Complete the square:

-7x^2+14x-21=0 move constant to other side, add 21 to both sides

-7x^2+14x=21 make the coefficient of x^2=1, divide both sides by -7

x^2-2x=-3 halve the linear coefficient, square it, add it to both sides, +1

x^2-2x+1=-2 now the left side is a perfect square

(x-1)^2=-2 take the square root of both sides

x-1=±√-2 add 1 to both sides

x=1±√-2 note that there are not real roots here because there is no real square root of a negative number...

x=1+i√2 and 1-i√2 so

(x+1+i√2)(x+1-i√2)

Answer 2

We can take out 7 because all of the terms include it:
7(-x^2+2x-3)
Take out negative 1:
-7(x^2-2x+3)

We cannot factor any further so the final answer is:
-7(x^2-2x+3)