consider a simpler case:
x^5 + 2x^3 + 5x^2
the common factor is x^2, that is, the power with the smallest exponent, so
x^2(x^3 + 2x + 5)
how did we get the terms inside ?
We subtracted the exponent of the common factor from the original exponent.
now to our question,
clearly the base of (x+4) becomes part of the common factor,
what is the smallest exponent ? it is -3/5
so (x+4)^(-3/5) is the common factor
(x+4)^(-3/5)[1 + (x+4)^(2/5) + (x+4)^(4/5)]
you can check this answer by expanding it, remember when multiplying powers we add the exponents
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