What is the following product? Assume xgeqslant 0 (4xsqrt (5x^2)+2x^2sqrt (6))^2 104x^4+8x^4sqrt (30x) 80x^6+8x^5+8x^5sqrt (30)+24x^4 104x^6 104x^4+16x^4sqrt (30)

The correct answer is $104x^{4}+16x^{4}\sqrt {30}$.<br /><br />To find the product, we need to expand the expression $(4x\sqrt {5x^{2}}+2x^{2}\sqrt {6})^{2}$.<br /><br />Expanding the expression, we get:<br /><br />$(4x\sqrt {5x^{2}}+2x^{2}\sqrt {6})^{2} = (4x\sqrt {5x^{2}})^{2} + 2(4x\sqrt {5x^{2}})(2x^{2}\sqrt {6}) + (2x^{2}\sqrt {6})^{2}$<br /><br />Simplifying each term, we have:<br /><br />$(4x\sqrt {5x^{2}})^{2} = 16x^{2}(5x^{2}) = 80x^{4}$<br /><br />$2(4x\sqrt {5x^{2}})(2x^{2}\sqrt {6}) = 16x^{3}\sqrt {30x}$<br /><br />$(2x^{2}\sqrt {6})^{2} = 4x^{4}(6) = 24x^{4}$<br /><br />Adding up all the terms, we get:<br /><br />$80x^{4} + 16x^{3}\sqrt {30x} + 24x^{4} = 104x^{4} + 16x^{3}\sqrt {30x}$<br /><br />Therefore, the product is $104x^{4}+16x^{4}\sqrt {30}$.