we have
[tex] 36x^{3} - 22x^{2} - 144x [/tex]
Step [tex] 1 [/tex]
Find the GCF of the polynomial
we know that
[tex] 36=2^{2} *3^{2} \\ Factors=1,2,3,4,6,9,12,18,36 [/tex]
[tex] x^{3}\\ Factors=1,x,x^{2},x^{3} [/tex]
[tex] 22=2*11\\ Factors=1,2,11,22 [/tex]
[tex] x^{2}\\ Factors=1,x,x^{2} [/tex]
[tex] 144=2^{4} *3^{2} \\ Factors=1,2,3,4,6,9,12,18,24,36,48,72,144 [/tex]
[tex] x\\ Factors=1,x [/tex]
So
[tex] GCF=2x [/tex]
[tex] 36x^{3} - 22x^{2} - 144x=2x*(18x^{2} - 11x - 72) [/tex]
analyze each case
case 1) [tex] 11 [/tex]
[tex] 11=1*11\\ Factors=1,11 [/tex]
the new GCF will be different [tex] GCF=1 [/tex]
[tex] 36x^{3} - 22x^{2} - 144x+11=(1)*(36x^{3} - 22x^{2} - 144x+11) [/tex]
case 2) [tex] 50xy [/tex]
[tex] 50=2*5^{2}\\ Factors=1,2,25,50 [/tex]
[tex] xy\\ Factors=1,x,y,xy [/tex]
the new GCF will be the same [tex] GCF=2x [/tex]
[tex] 36x^{3} - 22x^{2} - 144x+50xy=(2x)*(18x^{2} - 11x - 72+25y) [/tex]
case 3) [tex] 40x^{2} [/tex]
[tex] 40=2^{3}*5\\ Factors=1,2,8,10,20,40 [/tex]
[tex] x^{2}\\ Factors=1,x,x^{2} [/tex]
the new GCF will be the same [tex] GCF=2x [/tex]
[tex] 36x^{3} - 22x^{2} - 144x+40x^{2}=(2x)*(18x^{2} - 11x - 72+20x) [/tex]
case 4) [tex] 24 [/tex]
[tex] 24=2^{3}*3\\ Factors=1,2,4,6,12,24 [/tex]
the new GCF will be different [tex] GCF=2 [/tex]
[tex] 36x^{3} - 22x^{2} - 144x+24=(2)*(18x^{3} - 11x^{2} - 72x+12) [/tex]
case 5) [tex] 10y [/tex]
[tex] 10=2*5\\ Factors=1,2,5,10 [/tex]
[tex] y\\ Factors=1,y [/tex]
the new GCF will be different [tex] GCF=2 [/tex]
[tex] 36x^{3} - 22x^{2} - 144x+10y=(2)*(18x^{3} - 11x^{2} - 72x+5y) [/tex]
therefore
the answer is
[tex] 50xy [/tex]
[tex] 40x^{2} [/tex]
