[tex]a) x^{3}+4x ^2+x+4 = 4(x^{2} +1) + x(x^{2} +1) = (4+x)(x^2+1) \\b) 6x^3 - 5x^2+6x-5 = 6x(x^2+1) -5(x^2+1) = (6x-5)(x^2+1) \\c) x^3-\frac{1}{2}x^2+x-\frac{1}{2} = x^2(x-\frac{1}{2}) +1(x-\frac{1}{2}) = (x^2+1)(x-\frac{1}{2})\\d) x^3-5x^2+3x-15= 3(x-5) + x^2(x-5) = (x-5)(3+x^2) \\e) 2x^3-3x^2-6x+9 = 2x(-3+x^2) -3 (-3+x^2) = (2x-3)(x^2-3) \\f) 10x^3+25x^2-8x-20= 5x^2(2x+5) - 4(2x+5)= (5x^2-4)(2x+5) \\g) x^5+10x^4+x^3+10x^2= x^2(x^3+10x^2+x+10) = x^2 (10(x^2+1)+x(x^2+1)) = x^2(x^2+1)(x+10)\\[/tex]
[tex]h)3x^4-7x^3+3x^2-7x = 3x^2(x^2+1) -7x(x^2+1) = (3x^2-7x)(x^2+1)\\i)x^6+3x^5+2x^4+6x^3= x^3(x^3+3x^2+2x+6) = x^3( x^2(x+3) + 2(x+3)) = x^3(x+3)(x^2+2) \\j) 2x^5+5x^4+8x^3+20x^2= x^2(2x^3+5x^2+8x+20) = x^2( 5(x^2+4) +2x(x^2+4)) = x^2(5+2x)(x^2+4) \\k) 15x^6-10x^5+45x^4-30x^3= x^3(15x^3-10x^2+45x-30) =x^3(15(3x-2)+5x^2(3x-2))= x^3(15+5x^2) (3x-2) \\l) -24x^4+120x^3+30x^2-150x= 2x(-12x^3+60x^2+15x-75) = 2x(12x^2 (-x+5) -15(-x+5) = 2x(12x^2-15)(-x+5)[/tex]
