Question

Aratati ca daca x este un numar natural atunci numarul (x²+4x+3)(x²+4x+5)+1 este produsul a 4 numere naturale mai mari ca 1

Answer 1

(x²+4x+3)(x²+4x+5)+1=
=(x⁴+4x³+3x²+4x³+16x²+12x+5x²+20x+15+1=
=x⁴+8x³+24x²+32x+16=
=x⁴+6x³+2x³+12x²+12x²+8x+24x+16=
=(x⁴+2x³)+(6x³+12x²)+(12x²+24x)+(8x+16)=
=x³(x+2)+6x²(x+2)+12x(x+2)+8(x+2)=
=(x+2)(x³+6x²+12x+8)=
=(x+2)(x³+4x²+2x²+4x+8x+8)=
=(x+2)[x²(x+2)+4x(x+2)+4(x+2)]=
=(x+2)(x+2)(x²+4x+4)=
=(x+2)(x+2)(x+2)(x+2)=(x+2)⁴
x numar natural
(x+2)⁴>1