Multiply polynomial $ \, \color{blue}{ 1+4i } \, $ by $ \, \color{orangered}{ 1-4i }$.

$$ \left( \color{blue}{1+4i} \right) \cdot \left( \color{orangered}{1-4i} \right) = -16i^2+1 $$

Step 1: First we have to write polynomials in descending order.

$$ \begin{aligned} P(x) &= 4i+1 \\ Q(x) &= -4i+1 \\ \end{aligned} $$

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

$$ \begin{aligned} \left( \color{blue}{ 4i+1}\right) \cdot \left( \color{orangered}{ -4i+1}\right) &= \underbrace{ \color{blue}{4i} \cdot \left( \color{orangered}{-4i} \right) }_{\text{FIRST}} + \underbrace{ \color{blue}{4i} \cdot \color{orangered}{1} }_{\text{OUTER}} + \underbrace{ \color{blue}{1} \cdot \left( \color{orangered}{-4i} \right) }_{\text{INNER}} + \underbrace{ \color{blue}{1} \cdot \color{orangered}{1} }_{\text{LAST}} = \\ &= -16i^2 + 4i + \left( -4i\right) + 1 = \\ &= -16i^2 + 4i + \left( -4i\right) + 1 = \\ &= -16i^2+1; \end{aligned} $$

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