Multiply each term of $ \left( \color{blue}{x^4+3x^3-12x^2-20x+48}\right) $ by each term in $ \left( x^3+3x^2+3x+1\right) $.

$$ \left( \color{blue}{x^4+3x^3-12x^2-20x+48}\right) \cdot \left( x^3+3x^2+3x+1\right) = \\ = x^7+3x^6+3x^5+x^4+3x^6+9x^5+9x^4+3x^3-12x^5-36x^4-36x^3-12x^2-20x^4-60x^3-60x^2-20x+48x^3+144x^2+144x+48 $$

Combine like terms:

$$ x^7+ \color{blue}{3x^6} + \color{red}{3x^5} + \color{green}{x^4} + \color{blue}{3x^6} + \color{orange}{9x^5} + \color{blue}{9x^4} + \color{red}{3x^3} \color{orange}{-12x^5} \color{green}{-36x^4} \color{orange}{-36x^3} \color{blue}{-12x^2} \color{green}{-20x^4} \color{red}{-60x^3} \color{green}{-60x^2} \color{orange}{-20x} + \color{red}{48x^3} + \color{green}{144x^2} + \color{orange}{144x} +48 = \\ = x^7+ \color{blue}{6x^6} \color{green}{-46x^4} \color{red}{-45x^3} + \color{green}{72x^2} + \color{orange}{124x} +48 $$