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