\(a,x^3-5x^2+x-5=x^2\left(x-5\right)+\left(x-5\right)\\ =\left(x-5\right)\left(x^2+1\right)\\ \Rightarrow M=x^2+1\\ b,2x^4-13x^3+14x^2+15x=x\left(2x^3-13x^2+14x+15\right)\\ =x\left(2x^3-5x^2-8x^2+20x-6x+15\right)\\ =x\left(2x-5\right)\left(x^2-4x-3\right)\\ \Rightarrow M=x\left(2x-5\right)\\ d,2x^6-x^4-2x^2+1=x^4\left(2x^2-1\right)-\left(2x^2-1\right)\\ =\left(x^4-1\right)\left(2x^2-1\right)\\ \Rightarrow M=x^4-1=\left(x^2+1\right)\left(x-1\right)\left(x+1\right)\)
a) \(\Rightarrow x^2\left(x-5\right)+\left(x-5\right)=\left(x-5\right).M\)
\(\Rightarrow\left(x-5\right)\left(x^2+1\right)=\left(x-5\right).M\)
\(\Rightarrow M=x^2+1\)
b) \(\Rightarrow\left(x^2-4x-3\right).M=2x^2\left(x^2-4x-3\right)-5x\left(x^2-4x-3\right)\)
\(\Rightarrow\left(x^2-4x-3\right).M=\left(x^2-4x-3\right)\left(2x^2-5x\right)\)
\(\Rightarrow M=2x^2-5x\)
c) \(\Rightarrow\left(x^2+x+1\right).M=x^2\left(x^2+x+1\right)-2x\left(x^2+x+1\right)-3\left(x^2+x+1\right)\)
\(\Rightarrow\left(x^2+x+1\right).M=\left(x^2+x+1\right)\left(x^2-2x-3\right)\)
\(\Rightarrow M=x^2-2x-3\)
d) \(\Rightarrow x^4\left(2x^2-1\right)-\left(2x^2-1\right)=M\left(2x^2-1\right)\)
\(\Rightarrow\left(2x^2-1\right).\left(x^4-1\right)=M\left(2x^2-1\right)\)
\(\Rightarrow M=x^4-1\)
