How do I solve this particular problem, 5x^2 times 2y^4 time 3x^6 ?

You mean simplify?

When you have only multiplication, you can multiply them all together and put them in a single chunk.

First, work out all the numbers, then work out all the variables.

What is 5 x 2 x 3 ?

Then, what is x^2\ \times\ y^4 \ \times\ x^6 ?

If the base is the same, add the powers.

Can you post what you get?