\[ \left(4 x^{4}+3 y^{5}\right)\left(2 x^{4}-4 y^{3}\right) \] pand and simplify:

To expand the expression \((4x^{4}+3y^{5})(2x^{4}-4y^{3})\), we can use the distributive property, also known as the FOIL method for binomials. First, we’ll distribute \(4x^{4}\) over both terms in the second bracket: \[ 4x^{4} \cdot 2x^{4} = 8x^{8} \] \[ 4x^{4} \cdot (-4y^{3}) = -16x^{4}y^{3} \] Next, we distribute \(3y^{5}\) over both terms in the second bracket: \[ 3y^{5} \cdot 2x^{4} = 6x^{4}y^{5} \] \[ 3y^{5} \cdot (-4y^{3}) = -12y^{8} \] Now, we combine all these results: \[ 8x^{8} - 16x^{4}y^{3} + 6x^{4}y^{5} - 12y^{8} \] This yields the final simplified expression: \[ 8x^{8} - 16x^{4}y^{3} + 6x^{4}y^{5} - 12y^{8} \] And there you have it, fully expanded and simplified!