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The problem is 5x to the 3rd times (7x)squared.

I multiplied (7x)(7x) first and got x squared + 14x + 49. Then I multiplied each of those by 5x to the 3rd and came up with an answer of: 5x to the 5th + 70x to the 4th + 245x to the 3rd. Did I get this one correct??

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<<I multiplied (7x)(7x) first and got x squared + 14x + 49.>>

Well you did that wrong, as well as some other steps.
(7x)^2 = 49x^2
The polynomial you wrote is (x+7)^2

5x^3*(7x)^2 = 5*49 x^5 = 245 x^5
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