Multiply each term of $ \left( \color{blue}{x+y+1}\right) $ by each term in $ \left( x+y+1\right) $.

$$ \left( \color{blue}{x+y+1}\right) \cdot \left( x+y+1\right) = x^2+xy+x+xy+y^2+y+x+y+1 $$

Combine like terms:

$$ x^2+ \color{blue}{xy} + \color{red}{x} + \color{blue}{xy} +y^2+ \color{green}{y} + \color{red}{x} + \color{green}{y} +1 = x^2+ \color{blue}{2xy} +y^2+ \color{red}{2x} + \color{green}{2y} +1 $$

Multiply each term of $ \left( \color{blue}{x^2+2xy+y^2+2x+2y+1}\right) $ by each term in $ \left( x+y+1\right) $.

$$ \left( \color{blue}{x^2+2xy+y^2+2x+2y+1}\right) \cdot \left( x+y+1\right) = \\ = x^3+x^2y+x^2+2x^2y+2xy^2+2xy+xy^2+y^3+y^2+2x^2+2xy+2x+2xy+2y^2+2y+x+y+1 $$

Combine like terms:

$$ x^3+ \color{blue}{x^2y} + \color{red}{x^2} + \color{blue}{2x^2y} + \color{green}{2xy^2} + \color{orange}{2xy} + \color{green}{xy^2} +y^3+ \color{blue}{y^2} + \color{red}{2x^2} + \color{red}{2xy} + \color{green}{2x} + \color{red}{2xy} + \color{blue}{2y^2} + \color{orange}{2y} + \color{green}{x} + \color{orange}{y} +1 = \\ = x^3+ \color{blue}{3x^2y} + \color{green}{3xy^2} +y^3+ \color{red}{3x^2} + \color{red}{6xy} + \color{blue}{3y^2} + \color{green}{3x} + \color{orange}{3y} +1 $$