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

$$ \begin{aligned} P(x) &= 4x^3+3x \\ Q(x) &= -12x^4+25x^2 \\ \end{aligned} $$

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

$$ \begin{aligned} \left( \color{blue}{ 4x^3+3x}\right) \cdot \left( \color{orangered}{ -12x^4+25x^2}\right) &= \underbrace{ \color{blue}{4x^3} \cdot \left( \color{orangered}{-12x^4} \right) }_{\text{FIRST}} + \underbrace{ \color{blue}{4x^3} \cdot \color{orangered}{25x^2} }_{\text{OUTER}} + \underbrace{ \color{blue}{3x} \cdot \left( \color{orangered}{-12x^4} \right) }_{\text{INNER}} + \underbrace{ \color{blue}{3x} \cdot \color{orangered}{25x^2} }_{\text{LAST}} = \\ &= -48x^7 + 100x^5 + \left( -36x^5\right) + 75x^3 = \\ &= -48x^7 + 100x^5 + \left( -36x^5\right) + 75x^3 = \\ &= -48x^7+64x^5+75x^3; \end{aligned} $$

Polynomial Operations Calculator