Multiply each term of $ \left( \color{blue}{2x+3}\right) $ by each term in $ \left( x+1\right) $.

$$ \left( \color{blue}{2x+3}\right) \cdot \left( x+1\right) = 2x^2+2x+3x+3 $$

Combine like terms:

$$ 2x^2+ \color{blue}{2x} + \color{blue}{3x} +3 = 2x^2+ \color{blue}{5x} +3 $$

Multiply each term of $ \left( \color{blue}{2x^2+5x+3}\right) $ by each term in $ \left( x-2\right) $.

$$ \left( \color{blue}{2x^2+5x+3}\right) \cdot \left( x-2\right) = 2x^3-4x^2+5x^2-10x+3x-6 $$

Combine like terms:

$$ 2x^3 \color{blue}{-4x^2} + \color{blue}{5x^2} \color{red}{-10x} + \color{red}{3x} -6 = 2x^3+ \color{blue}{x^2} \color{red}{-7x} -6 $$

Multiply each term of $ \left( \color{blue}{2x^3+x^2-7x-6}\right) $ by each term in $ \left( x^2+3\right) $.

$$ \left( \color{blue}{2x^3+x^2-7x-6}\right) \cdot \left( x^2+3\right) = 2x^5+6x^3+x^4+3x^2-7x^3-21x-6x^2-18 $$

Combine like terms:

$$ 2x^5+ \color{blue}{6x^3} +x^4+ \color{red}{3x^2} \color{blue}{-7x^3} -21x \color{red}{-6x^2} -18 = 2x^5+x^4 \color{blue}{-x^3} \color{red}{-3x^2} -21x-18 $$