find the first 3 terms in the expansion of (x+4)(1+3x)^-2 as a series in ascending powers.

I expanded the second bracket using binomial expansion and got 1 -6x +27x^2 -108x^3

how do I combine (x+4) with the expansion?

Any help would be appreciated.

just multiply out as usual, taking only the firet 3 terms:

(1 -6x +27x^2 -108x^3)(4+x)
= (1 -6x +27x^2 -108x^3)*4
+ (1 -6x +27x^2 -108x^3)*x

= 4 - 24x + 108x^2 - ...
+ x - 6x^2 + 27x^3 - ...
----------------------------------
4 - 23x + 102x^2 + ...

To combine (x+4) with the expanded series of (1+3x)^-2, you need to multiply each term in the expansion by (x+4). Let's go through the process step by step:

The expansion of (1+3x)^-2 using binomial expansion is:
1 - 6x + 27x^2 - 108x^3 + ...

To combine it with (x+4), you will multiply each term in the expansion by (x+4):

For the first term: (1)(x+4) = x+4
For the second term: (-6x)(x+4) = -6x^2 - 24x
For the third term: (27x^2)(x+4) = 27x^3 + 108x^2
For the fourth term: (-108x^3)(x+4) = -108x^4 - 432x^3

So, the combined series in ascending powers is:
x+4 - 6x^2 - 24x + 27x^3 + 108x^2 - 108x^4 - 432x^3 + ...

Now, you can simplify and rearrange this series as needed. The first three terms in the expansion are:
x + 4 - 6x^2 - 24x

I hope this helps! Let me know if you have any further questions.