Answer:
The values of k will be:
[tex]k=-1,\:k=3[/tex]
Step-by-step explanation:
Let the expression of polynomial P be
[tex]P\left(x\right)=3x^2-4kx-4k^2[/tex]
Let the expression if the polynomial Q be
[tex]\:Q\left(x\right)=\:\left(x+2\right)\:[/tex]
Plug in Q(x) = 0
0 = x+2
x = -2
As (x+2) is a factor of 3x²-4kx-4k²
substitute x = -2 in the the polynomial
3x²-4kx-4k² = 0
[tex]3\left(-2\right)^2-4k\left(-2\right)-4k^2\:=0[/tex]
[tex]12+8k-4k^2=0[/tex]
Write in the standard form ax²+bx+c = 0
[tex]-4k^2+8k+12=0[/tex]
Factor out common term -4
[tex]-4\left(k^2-2k-3\right)=0[/tex]
Factor k²-2k-3: (k+1)(k-3)
[tex]-4\left(k+1\right)\left(k-3\right)=0[/tex]
Using the zero factor principle
if ab=0, then a=0 or b=0 (or both a=0 and b=0)
[tex]k+1=0\quad \mathrm{or}\quad \:k-3=0[/tex]
solving k+1=0
k+1 = 0
k = -1
solving k-3=0
k-3=0
k = 3
Thus, the values of k will be:
[tex]k=-1,\:k=3[/tex]
