Answer:
The simplified product is:
[tex]10x^4\sqrt{6}+x^3\sqrt{30x}-10x^4\sqrt{3}-x^3\sqrt{15x}[/tex]
Step-by-step explanation:
The expression is given by:
[tex](\sqrt{10x^4}-x\sqrt{5x^2})(2\sqrt{15x^4}+\sqrt{3x^3})[/tex]
Now we know that:
[tex]\sqrt{x^2}=x\\\\and\\\\\sqrt{x^4}=\sqrt{(x^2)^2}=x^2[/tex]
Hence, we get the expression as follows:
[tex](\sqrt{10x^4}-x\sqrt{5x^2})(2\sqrt{15x^4}+\sqrt{3x^3})=(x^2\sqrt{10}-x\cdot x\sqrt{5})(2x^2\sqrt{15}+x\sqrt{3x})[/tex]
Now, we will use the property that:
[tex](a+b)(c+d)=a(c+d)+b(c+d)[/tex]
Hence, we have the expression as:
[tex]=x^2\sqrt{10}(2x^2\sqrt{15}+x\sqrt{3x})-x^2\sqrt{5}(2x^2\sqrt{15}+x\sqrt{3x})[/tex]
[tex]=2x^4\sqrt{10}\sqrt{15}+x^3\sqrt{10}\sqrt{3x}-2x^4\sqrt{5}\sqrt{15}-x^3\sqrt{5}\sqrt{3x}[/tex]
Now we know that:
[tex]\sqrt{a}\sqrt{b}=\sqrt{ab}[/tex]
i.e. we have:
[tex]=2x^4\sqrt{150}+x^3\sqrt{30x}-2x^4\sqrt{75}-x^3\sqrt{15x}\\\\=2x^4\sqrt{5^2\cdot 6}+x^3\sqrt{30x}-2x^4\sqrt{5^2\cdot 3}-x^3\sqrt{15x}\\\\=10x^4\sqrt{6}+x^3\sqrt{30x}-10x^4\sqrt{3}-x^3\sqrt{15x}[/tex]
