<s>[INST] Does $4(x^3+2x^2+x)^3(3x^2+4x+1) = 4x^3(3x+1)(x+1)^7$? Please provide proof
The answer in the back of the book is different to both my calculations and also the online calculator I crosschecked my answer with...

The Question:
"Differentiate with respect to $x$:"
$
(x^3+2x^2+x)^4
$
My Answer: $4(x^3+2x^2+x)^3(3x^2+4x+1)$
The Book's Answer: $4x^3(3x+1)(x+1)^7$
 [/INST] You're both right.
Note that
$$
x^3+2x^2+x = x (x + 1)^2
$$
$$
3x^2+4x+1 = (3 x + 1) (x + 1)
$$
On the other hand, you might have started with
$$
(x^3+2x^2+x)^4 = x^4 (x + 1)^8
$$
whose derivative is
$$
4x^3 (x + 1)^8 + 8x^4 (x + 1)^7
= 4x^3 (x+1 +2x) (x + 1)^7
= 4x^3 (3x+1) (x + 1)^7
$$
</s>