\(a^3+b^3-\left(a^2-2ab+b^2\right)\left(a-b\right)=a^3+b^3-\left(a-b\right)^3\)

\(=a^3+b^3-\left(a^3-3a^2b+3ab^2-b^3\right)\)

\(=a^3+b^3-a^3+3a^2b-3ab^2+b^3=3a^2b-3ab^2+2b^3\)

`a^3 + b^3 - (a^2 - 2ab + b^2) (a-b)`

`= a^3 + b^3 - (a-b)^2 (a-b)`

`= a^3 + b^3 - (a-b)^3`

`= a^3 + b^3 - (a^3 - 3a^2b + 3ab^2 - b^3)`

`= a^3 + b^3 - a^3 +3a^2b - 3ab^2 +b^3`

`= 3a^2b - 3ab^2 + 2b^3`

\(\left(\frac{5}{3}x+y\right)\left(y^2+y+3\right)=\frac{5}{3}x\left(y^2+y+3\right)+y\left(y^2+y+3\right)=\frac{5}{3}xy^2+\frac{5}{3}xy+5x+y^3+y^2+3y\)

`(5/3x +y) (y^2 +y+3)`

`= 5/3 x (y^2 +y+3)+y (y^2 +y+3)`

`= 5/3x . y^2 + 5/3x . y + 5/3 x.3 + y . y^2 + y.y +3.y`

`= 5/3 xy^2 + 5/3 xy + 5x + y^3+y^2 +3y`

(4x - 5)^2 = (4x)^2 - 2.4x.(-5) + (5)^2 = 16x^2 - 13x + 25 

(4x-5)^2

= ( 4x)^2  -  2 * 4x * 5 + 5^2

= 16x^2   - 40x   +  25

x^{n - 1} (x + y) - y (x^{n - 1} + y^{n - 1}) \(=x^n+x^{n-1}y-x^{n-1}y-y^n=x^n-y^n\)

x^n-1(x+y)-y(x^n-1+y^n-1)

= xⁿ+x^n-1y-yx^n-1-yⁿ

=xⁿ+(x^n-1y-yx^n-1)-yⁿ

=xⁿ-yⁿ

-13/2xy.(-x^2-xy-5)=-13xy.(-x^2)+(-13/2).(-xy)+(-13/2).(-5)=13x^3y+13/2x^2y^265/2xy

`( (-13)/2xy) (-x^2 - xy - 5)`

`= (-13)/2 xy . (-x)^2 +(-13)/2xy . (-xy) + (-13)/2xy . (-5)`

`= 13/2 x^3y + 13/x^2y^2 + 65/2 xy`