Answer:
[tex] -(\lambda -4)(\lambda -2)(\lambda +2) [/tex]
Step-by-step explanation:
What we want to fator is:
[tex] -\lambda^3 + 4 \lambda^2 + 4\lambda - 16[/tex]
There is no common factor, but let's factor it by grouping. The first two addends can be factor as follows:
[tex]-\lambda^3 + 4 \lambda^2 = \lambda^2(-\lambda + 4) = -\lambda^2(\lambda - 4)[/tex]
the second addends can be factor as well:
[tex]4\lambda - 16 = 4(\lambda- 4)[/tex].
Then our original expression can be rewritten like
[tex]-\lambda^3 + 4 \lambda^2 + 4\lambda -16=\lambda^2(\lambda - 4) + 4(\lambda - 4)[/tex]
And here the [tex](\lambda-4)[/tex] is the common factor!
[tex] -\lambda^2(\lambda - 4) + 4(\lambda - 4) = (\lambda - 4)(-\lambda^2 + 4)[/tex]
Finally, we can factor the quadratic expression as a difference of squares [tex] -\lambda^2 + 4 = 4 - \lambda^2 = (2+\lambda)(2-\lambda)[/tex]
Ant we get
[tex](\lambda - 4)(-\lambda^2 + 4)= (\lambda - 4)(\lambda + 2)(2-\lambda)[/tex]
now, we can extract the negative sign from [tex](2-\lambda)[/tex], and we get
[tex] -(\lambda -4)(\lambda -2)(\lambda +2) [/tex].
