Multiply polynomial $ \, \color{blue}{ x^3+1 } \, $ by $ \, \color{orangered}{ x-1 }$.

$$ \left( \color{blue}{x^3+1} \right) \cdot \left( \color{orangered}{x-1} \right) = x^4-x^3+x-1 $$

In this example we are multiplying two binomials so FOIL method can be used.

$$ \begin{aligned} \left( \color{blue}{ x^3+1}\right) \cdot \left( \color{orangered}{ x-1}\right) &= \underbrace{ \color{blue}{x^3} \cdot \color{orangered}{x} }_{\text{FIRST}} + \underbrace{ \color{blue}{x^3} \cdot \left( \color{orangered}{-1} \right) }_{\text{OUTER}} + \underbrace{ \color{blue}{1} \cdot \color{orangered}{x} }_{\text{INNER}} + \underbrace{ \color{blue}{1} \cdot \left( \color{orangered}{-1} \right) }_{\text{LAST}} = \\ &= x^4 + \left( -x^3\right) + x + \left( -1\right) = \\ &= x^4 + \left( -x^3\right) + x + \left( -1\right) = \\ &= x^4-x^3+x-1; \end{aligned} $$

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