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

$$ \begin{aligned} P(x) &= -x^8+x \\ Q(x) &= -x^5+x \\ \end{aligned} $$

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

$$ \begin{aligned} \left( \color{blue}{ -x^8+x}\right) \cdot \left( \color{orangered}{ -x^5+x}\right) &= \underbrace{ \left( \color{blue}{-x^8} \right) \cdot \left( \color{orangered}{-x^5} \right) }_{\text{FIRST}} + \underbrace{ \left( \color{blue}{-x^8} \right) \cdot \color{orangered}{x} }_{\text{OUTER}} + \underbrace{ \color{blue}{x} \cdot \left( \color{orangered}{-x^5} \right) }_{\text{INNER}} + \underbrace{ \color{blue}{x} \cdot \color{orangered}{x} }_{\text{LAST}} = \\ &= x^{13} + \left( -x^9\right) + \left( -x^6\right) + x^2 = \\ &= x^{13} + \left( -x^9\right) + \left( -x^6\right) + x^2 = \\ &= x^{13}-x^9-x^6+x^2; \end{aligned} $$

Polynomial Operations Calculator