Szczegółowe wyjaśnienie:
a) Skorzystamy ze wzoru:
[tex](a+b)^2=a^2+2ab+b^2[/tex]
[tex](x+2)^2+(x+2)^2=2(x+2)^2=2(x^2+2\cdot x\cdot 2+2^2)\\\\=2(x^2+4x+4)=2\cdot x^2+2\cdot4x+2\cdot4=2x^2+8x+8[/tex]
b) Skorzystamy ze wzorów:
[tex](a+b)^3=a^3+3a^2b+3ab^2+b^3\\(a-b)^2=a^2-2ab+b^2[/tex]
[tex]\left(x+\dfrac{2}{3}\right)^3-\left(x-\dfrac{2}{3}\right)^2\\\\=x^3+3\cdot x^2\cdot\dfrac{2}{3}+3\cdot x\cdot\left(\dfrac{2}{3}\right)^2+\left(\dfrac{2}{3}\right)^3-\bigg[x^2-3\cdot x\cdot\dfrac{2}{3}+\left(\dfrac{2}{3}\right)^2\bigg]\\\\=x^3+2x^2+\dfrac{4}{3}x+\dfrac{8}{27}-x^2+2x-\dfrac{4}{9}\\\\=x^3+x^2+3\dfrac{1}{3}x-\dfrac{4}{27}[/tex]
c) Skorzystamy ze wzorów:
[tex](a+b)^3=a^3+3a^2b+3ab^2+b^3\\(a-b)^3=a^3-3a^2b+3ab^2-b^3[/tex]
[tex](3x+2)^2-(3x-2)^3\\\\=(3x)^3+3\cdot(3x)^2\cdot2+3\cdot3x\cdot2^2+2^3-\bigg[(3x)^3-3\cdot(3x)^2\cdot2+3\cdot3x\cdot2^2-2^3\bigg]\\\\=27x^3+54x^2+36x+8-27x^3+54x^2-36x+8\\\\=108x^2+16[/tex]
d) Skorzystamy ze wzorów:
[tex](a-b)^3=a^3-3a^2b+3ab^2-b^3\\(a+b)^3=a^3+3a^2b+3ab^2+b^3[/tex]
[tex](x^2-2x)^3-(x^2+2x)^3\\\\=(x^2)^3-3\cdot (x^2)^2\cdot2x+3\cdot x^2\cdot(2x)^2-(2x)^3-\bigg[(x^2)^3+3\cdot (x^2)^2\cdot2x+3\cdot x^2\cdot(2x)^2+(2x)^3\bigg]\\\\=x^6-6x^5+12x^4-8x^3-x^6-6x^5-12x^4-8x^3\\\\=-16x^3[/tex]
