4.
a)
1. 1/(n²+n)=1/(n*(n+1).
2. 1/(n²+3n+2)=1/(n²+n+2n+2)=1/(n*(n+1)+2*(n+1))=1/((n+1)(n+2)).
3. 1/(n*(n+1)-1/((n+1)(n+2))=(n+2-n)/(n*(n+1)*(n+2))=2/(n*(n+1)(n+2)).
4. (n-3)/(n²+2n)=(n-3)/(n*(n+2)).
5. 2*(n*(n+2))/(n*(n+1)(n+2)(n-3)=2/((n+1)(n-3)).
б)
2/((0+1)(0-3))=2/(1*(-3))=-2/3.
5.
a)
х²+у²+4х-6у+13≥0
(x²+4x+4)+(y²-6x+9)≥0
(x+2)²+(y-3)²≡≥0.
б)
x⁴+10x²-4x+14>0
x⁴+9x²+(x²-4x+4)+10>0
x⁴+9x²+(x-2)²+10≡>0
в)
x²+4>√(x⁴+8x²+15)
(x²+4)²>(√(x⁴+8x²+15))²
x⁴+8x²+16>x⁴+8x²+15
1≡>0.
6.
x⁴-4x³-2x²+12x+9=0
x⁴+(2x³-6x³)+(x²-12x²+9x²)+(18x-6x)+9=0
(x⁴+2x²+x²)+(-6x³-12x²-6x)+(9x²+18x+9)=0
x²*(x²+2x+1)-6x*(x²+2x+1)+9*(x²+2x+1)=0
(x²-6x+9)*(x²+2x+1)=0
(x-3)²*(x+1)²=0
(x-3)²=0
x-3=0
x₁=3
(x+1)²=0
x+1=0
x₂=-1
Ответ: x₁=3 x₂=-1.