$\left( 4{x}^{4}-8{y}^{3} \right) \times \left( 4{x}^{4}-8{y}^{3} \right)$
Multiply each term in the first parentheses by each term in the second parentheses (FOIL)$4{x}^{4} \times 4{x}^{4}-4{x}^{4} \times 8{y}^{3}-8{y}^{3} \times 4{x}^{4}-8{y}^{3} \times \left( -8{y}^{3} \right)$
Calculate the product$16{x}^{8}-4{x}^{4} \times 8{y}^{3}-8{y}^{3} \times 4{x}^{4}-8{y}^{3} \times \left( -8{y}^{3} \right)$
Calculate the product$16{x}^{8}-32{x}^{4}{y}^{3}-8{y}^{3} \times 4{x}^{4}-8{y}^{3} \times \left( -8{y}^{3} \right)$
Calculate the product$16{x}^{8}-32{x}^{4}{y}^{3}-32{x}^{4}{y}^{3}-8{y}^{3} \times \left( -8{y}^{3} \right)$
Multiplying two negatives equals a positive: $\left( - \right) \times \left( - \right)=\left( + \right)$$16{x}^{8}-32{x}^{4}{y}^{3}-32{x}^{4}{y}^{3}+8{y}^{3} \times 8{y}^{3}$
Calculate the product$16{x}^{8}-32{x}^{4}{y}^{3}-32{x}^{4}{y}^{3}+64{y}^{6}$
Collect like terms$16{x}^{8}-64{x}^{4}{y}^{3}+64{y}^{6}$