Multiply polynomial $ \, \color{blue}{ 3x^2-4x } \, $ by $ \, \color{orangered}{ 4x^2+5x }$.

$$ \left( \color{blue}{3x^2-4x} \right) \cdot \left( \color{orangered}{4x^2+5x} \right) = 12x^4-x^3-20x^2 $$

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

$$ \begin{aligned} \left( \color{blue}{ 3x^2-4x}\right) \cdot \left( \color{orangered}{ 4x^2+5x}\right) &= \underbrace{ \color{blue}{3x^2} \cdot \color{orangered}{4x^2} }_{\text{FIRST}} + \underbrace{ \color{blue}{3x^2} \cdot \color{orangered}{5x} }_{\text{OUTER}} + \underbrace{ \left( \color{blue}{-4x} \right) \cdot \color{orangered}{4x^2} }_{\text{INNER}} + \underbrace{ \left( \color{blue}{-4x} \right) \cdot \color{orangered}{5x} }_{\text{LAST}} = \\ &= 12x^4 + 15x^3 + \left( -16x^3\right) + \left( -20x^2\right) = \\ &= 12x^4 + 15x^3 + \left( -16x^3\right) + \left( -20x^2\right) = \\ &= 12x^4-x^3-20x^2; \end{aligned} $$

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