Question

Factor -4x^2+16x-24 completely

Answer 1

-4x²+16x-24=-1(4x²-16x+24)=-4(x²-4x+6)
x²-4x+6 has no real roots. ☺☺☺☺

Answer 2

Let's complete the square (as this is actually how the quadratic formula is derived)

-4x^2+16x-24=0  make the leading coefficient equal to 1 by dividing the entire equation by -4

x^2-4x+6=0, move the constant to the other side by subtracting 6 from both sides

x^2-4x=-6, halve the linear coefficient, square it, then add it to both sides, (-4/2)^2=4

x^2-4x+4=-2  now the left side is a perfect square

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

x-2=±i√2  note the imaginary unit i as there is no real square root of -2, now add 2 to both sides

x=2±i√2 so the factored quadratic is:

(x+2-i√2)(x+2+i√2)

Note that these are not real factors, if you were to graph the original equation, it will never intersect with the x-axis...