Use the radical multiplication and simplification rules:
Multiplication: sqrta*sqrtb=sqrt(ab)
Simplification: sqrt(a^2)=a
For this problem, first, multiply the radicals (in blue) and their coefficients (in red) together:
color(white)=color(red)xcolor(blue)sqrt(10x)*color(red)7color(blue)sqrt(15x)
=color(red)x*color(blue)sqrt(10x)*color(red)7*color(blue)sqrt(15x)
=color(red)x*color(red)7*color(blue)sqrt(10x)*color(blue)sqrt(15x)
=color(red)(7x)*color(blue)sqrt(10x)*color(blue)sqrt(15x)
=color(red)(7x)*color(blue)sqrt(10x*15x)
=color(red)(7x)*color(blue)sqrt(10*15*x*x)
=color(red)(7x)\*color(blue)sqrt(150*x*x)
=color(red)(7x)*color(blue)sqrt(150*x^2)
Next, use the multiplication rule backwards:
color(white)=color(red)(7x)*color(blue)sqrt(150*x^2)
=color(red)(7x)*color(blue)sqrt(150)*color(blue)(sqrt(x^2))
Now, use the simplification rule:
color(white)=color(red)(7x)*color(blue)sqrt(150)*color(blue)(sqrt(x^2))
=color(red)(7x)*color(blue)sqrt(150)*color(red)x
=color(red)(7x)*color(red)x*color(blue)sqrt(150)
=color(red)(7x^2)*color(blue)sqrt(150)
Technically, this answer is correct, but it can be simplified further by factoring 150 and then using the simplification rule backward again:
color(white)=color(red)(7x^2)*color(blue)sqrt(150)
=color(red)(7x^2)*color(blue)sqrt(6*25)
=color(red)(7x^2)*color(blue)sqrt6*color(blue)sqrt25
=color(red)(7x^2)*color(blue)sqrt6*color(blue)sqrt(5^2)
=color(red)(7x^2)*color(blue)sqrt6*color(red)5
=color(red)(7x^2)*color(red)5*color(blue)sqrt6
=color(red)(35x^2)*color(blue)sqrt6
This answer is fully simplified.