I am trying to figure out how to factor out any common factors out of this polynomial: 4(x^{2}+4)^{3/2}(3x+5)^{1/3}+ (3x+5)^{4/3}(x^{2}+4)^{1/2}3x

What I thought to do was to look for each base raised to the smallest exponent present in each term. Looking at the polynomial it looks like (3x+5)^{1/3}, and (x^{2}+4)^{1/2}are the bases raised to the lowest exponent because 3/2 > 1/2, and 4/3>1/3 therefore I tried to factor these out. When I try to do this I end up with 4(x^{2}+4)^{2/2}+(3x+5)^{3/3}3x[(3x+5)(x^{2}+4)], and when I multiply the two binomials in the brackets I get 3x^{3}+5x^{2}+12x+20.

The answer in the text is (3x+5)^{1/3}(x^{2}+4)^{1/2}(13x^{2}+15x+16). I don't see how they arrived at this, and I also don't see how you could arrive at the original polynomial through multiplication of the terms in the answer. If someone could tell me what I am doing wrong, and explain how to arrive at the correct answer I would really appreciate it. Thanks a lot, I love this site.