Multiply polynomial $ \, \color{blue}{ 5x^6+2x^5-4x^4+ \cancel{3x^3}+6x^2-x+7 -\cancel{3x^3}+2x^2+4x-2 } \, $ by $ \, \color{orangered}{ 4x^3 }$.

$$ \left( \color{blue}{5x^6+2x^5-4x^4+ \cancel{3x^3}+6x^2-x+7 -\cancel{3x^3}+2x^2+4x-2} \right) \cdot \color{orangered}{4x^3} = 20x^9+8x^8-16x^7+32x^5+12x^4+20x^3 $$

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

$$ \begin{aligned} P(x) &= 5x^6+2x^5-4x^4+8x^2+3x+5 \\ Q(x) &= 4x^3 \\ \end{aligned} $$

We can use distribution property to multiply the polynomial by the monomial.

$$ \begin{aligned} \left( 5x^6+2x^5-4x^4+8x^2+3x+5 \right) \cdot \color{orangered}{4x^3} &= 5x^6 \cdot \color{orangered}{4x^3} + 2x^5 \cdot \color{orangered}{4x^3} + \left( -4x^4\right) \cdot \color{orangered}{4x^3} + 8x^2 \cdot \color{orangered}{4x^3} + 3x \cdot \color{orangered}{4x^3} + 5 \cdot \color{orangered}{4x^3} = \\ &= 20x^9+8x^8-16x^7+32x^5+12x^4+20x^3 \end{aligned} $$

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