start with the given equation

`r(x) = x^3 + x + 1 `

notice how we may plug values into the equation for x!

**example:**

`r(1) = 1 + 1 + 1 = 3`

now instead of plugging in real values lets plug in expressions

**start with r(x + 1)**, simply plug x + 1 into our equation

`r(x + 1) = (x+1)^3 + (x + 1) + 1 = (x + 1)^3 + x + 2`

**now plug in r(x^2), **once again plug into original equation

`r(x^2) = (x^2)^3 + x^2 + 1 = x^6 + x^2 + 1 `

solve for `r(x+1) - r(x^2) `

`((x+1)^3 + x + 2) - (x^6 + x^2 + 1) = r(x+1) - r(x^2)`

*you can keep it like this, it is correct. However we may simplify this also*

`((x^2 + 2x + 1)(x+1) + x + 2) - (x^6 + x^2 + 1)`

`(x^3 + 2x^2 + x + x^2 + 2x + 1 + x + 2) - (x^6 + x^2 + 1)`

`(x^3 + 3x^2 + 4x + 3) - (x^6 + x^2 + 1) `

**`-x^6 +x^3 +2x^2 + 4x + 2 = r(x + 1) - r(x^2)` **

*just the simplified version*