How to Deduce the Coefficient of x^n in the Expansion of (x+3)(1+2x)^(-2)?

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To deduce the coefficient of x^n in the expansion of (x+3)(1+2x)^(-2), the first step is to expand (1+2x)^(-2) to find its first four terms, which are 1 - 4x + 12x^2 - 32x^3. The coefficient of x^n in this expansion can be expressed as [(-1)^n](n+1)(2^n). The critical insight for the final part is recognizing that the only way to obtain an x term in the product is by multiplying the x term from (x+3) with the corresponding x^n term from the expansion. This method allows for the deduction of the coefficient of x^n in the overall expression.
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Homework Statement


By finding the first 4 terms in the expansion of (1+2x)^(-2) in ascending powers of x and find the coefficient of x^n term. Hence deduce the coefficient of x^n term in the expansion of (x+3)(1+2x)^(-2).


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The Attempt at a Solution


The first two parts were easy. For the first part, the answer is 1 - 4x + 12x^2 - 32x^3 + ...
For the second part, [(-1)^n](n+1)(2^n). But I'm stunned at the last part. How do I deduce?
 
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Well the only term that will give you an 'x' term is if you multiply an x by an x. So if you multiply the xn term by the x, you will get the nth term for the expansion.
 

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