To illustrate what I mean, consider the expression
$$ −A^4 + 4 A^3 B − 4 A^3 D − 6 A^2 B^2 + 12 A^2 B D − 6 A^2 D^2 + 4 A B^3 − 12 A B^2 D + 12 A B D^2 − 4 A D^3 $$
This can be re-written much more compactly as
$$ (B − D)^4 − (A − B + D)^4 $$
I started with the first expression and, through much guesswork and trial-and-error experimentation, found the second expression. I have multiple other, longer expressions that I want to simplify in a similar way, but to do so with the same process feels quite daunting. For example, two of the other expressions are
$$ A^4 − 4 A^3 B + 18 A^2 B^2 − 12 A^2 B G − 12 A^2 B H + 20 A B^3 + 24 A B^2 D − 36 A B^2 G − 36 A B^2 H − 24 A B D G − 24 A B D H + 12 A B G^2 + 24 A B G H + 12 A B H^2 + 12 B^4 + 24 B^3 D − 24 B^3 G − 24 B^3 H + 12 B^2 D^2 − 36 B^2 D G − 36 B^2 D H + 12 B^2 G^2 + 36 B^2 G H + 12 B^2 H^2 − 12 B D^2 G − 12 B D^2 H + 12 B D G^2 + 24 B D G H + 12 B D H^2 − 12 B G^2 H − 12 B G H^2 $$
and
$$ 4 A^3 H + 12 A^2 B H + 12 A^2 D H − 12 A^2 G H − 6 A^2 H^2 + 12 A B^2 H − 24 A B G H − 12 A B H^2 + 12 A D^2 H − 24 A D G H − 12 A D H^2 + 12 A G^2 H + 12 A G H^2 + 4 A H^3 + 4 B^3 H + 12 B^2 D H − 12 B^2 G H − 6 B^2 H^2 + 12 B D^2 H − 24 B D G H − 12 B D H^2 + 12 B G^2 H + 12 B G H^2 + 4 B H^3 + 4 D^3 H − 12 D^2 G H − 6 D^2 H^2 + 12 D G^2 H + 12 D G H^2 + 4 D H^3 − 4 G^3 H − 6 G^2 H^2 − 4 G H^3 − H^4 $$
I believe that these expression can be broken up into simple sums-and-or-differences of linear combinations of their variables raised to the fourth power, just like the expression at the start of this question, but I don't know of a good way to approach such long polynomials. Is there an efficient algorithm I can use to do this by hand? If not, is there a website or program that can compute it? At 379 and 405 characters long, respectively, those last expressions are way over the character limit for WolframAlpha.
For simplifying expressions, sometimes what to do is "depress" them by a change of variables (or get rid of intermediate terms just like for equations). When it's multi-variable expressions, that's when it becomes tricky.
I. First
$$ X = −A^4 + 4 A^3 B − 4 A^3 D − 6 A^2 B^2 + 12 A^2 B D − 6 A^2 D^2 + 4 A B^3 − 12 A B^2 D + 12 A B D^2 − 4 A D^3 $$
Let $D = B+C$, then,
$$X=-(A + C)^4 + C^4$$
Note that $B$ disappears, so the expression really had only two parameters.
II. Second
The second and third expressions have five variables $A,B,D,G,H$ and to express them as sums of fourth powers $x_i^4\pm(x_i\pm x_j)^4\pm(x_i\pm x_j\pm x_k)^4\pm(x_i\pm x_j\pm x_k\pm x_l)^4$ would be very tricky indeed. It was probably just luck the first was such. However, we can still simplify them.
$$\color{blue}Y = A^4 − 4 A^3 B + 18 A^2 B^2 − 12 A^2 B G − 12 A^2 B H + 20 A B^3 + 24 A B^2 D − 36 A B^2 G − 36 A B^2 H − 24 A B D G − 24 A B D H + 12 A B G^2 + 24 A B G H + 12 A B H^2 + 12 B^4 + 24 B^3 D − 24 B^3 G − 24 B^3 H + 12 B^2 D^2 − 36 B^2 D G − 36 B^2 D H + 12 B^2 G^2 + 36 B^2 G H + 12 B^2 H^2 − 12 B D^2 G − 12 B D^2 H + 12 B D G^2 + 24 B D G H + 12 B D H^2 − 12 B G^2 H − 12 B G H^2 $$
Let,
\begin{align} D &= \tfrac12(- B + P + Q - R) - A\\ G &= \tfrac12(+ B - P + Q - R)\\ H &= \tfrac12(+ B + P - Q - R) \end{align}
then we get the much simpler,
$$\color{blue}Y = (A - B)^4 - B^4 + 12 B P Q R$$
If $PQR=0$, then it is a difference of two fourth powers, like the first.
III. Third
The has a simplified form analogous to the second.
$$ \color{red}Z = 4 A^3 H + 12 A^2 B H + 12 A^2 D H − 12 A^2 G H − 6 A^2 H^2 + 12 A B^2 H − 24 A B G H − 12 A B H^2 + 12 A D^2 H − 24 A D G H − 12 A D H^2 + 12 A G^2 H + 12 A G H^2 + 4 A H^3 + 4 B^3 H + 12 B^2 D H − 12 B^2 G H − 6 B^2 H^2 + 12 B D^2 H − 24 B D G H − 12 B D H^2 + 12 B G^2 H + 12 B G H^2 + 4 B H^3 + 4 D^3 H − 12 D^2 G H − 6 D^2 H^2 + 12 D G^2 H + 12 D G H^2 + 4 D H^3 − 4 G^3 H − 6 G^2 H^2 − 4 G H^3 − H^4 $$
Let $G = A + B + D + J$, then it is just,
$$\color{red}Z = -(H+J)^4 +J^4 -24 A B D H $$
or explicitly,
$$\color{red}Z = -(A + B + D -G - H)^4 + (A + B + D -G)^4 -24 A B D H $$
If $ABD=0$, then it is also a difference of two fourth powers, like the first two.