I can't understand a development problem, I , please

[superduck]

Homework Statement

I'm learning mathematics in Japanese, so maybe my English is not correct to discribe what I want to mention.
I'm struggling a development probloem below

( x + y + 2z)³ – ( y + 2z – x)³ – ( 2z + x – y )³ – ( x + y – 2z )³

Homework Equations

I used cube development formula like a³ + b³= ( a+ b )³ -3ab( a + b)

The Attempt at a Solution

Homework Statement

I solved this problem, but the answer of my reference book used another way to solve it. Then, I can't understand that way.

My attempt at a solution is below

transpose　　 {x + y + 2z}³ - {　y＋2z - x }³=P,
{ 2z + x - y }³ + { x + y - 2z}³=Q

P=[{( y+2z) + x } - {( y + 2z) - x}]³ + 3{( y + 2z ) + x}{( y + 2Z) - x}{( y + 2z + x) - ( y + 2z - x)}
=(2x)³ + 3{( y + 2z )² - x²}　・2x
=8x³ + 6x( y² + 4yz + 4z² - x²)
=8x³ - 6x³ + 6xy² + 24xyz + 24z²x
=2x³ + 6xy² + 24z²x - 24xyzQ=[{ x -( y - 2z)} + { x + ( y - 2z)}]³ - 3{ x - ( y - 2z)}{ x + ( y - 2z)}×( x - y + 2z + x + y -2z)
=(2x)³ - 3{ x² - ( y - 2z)²}・2x
=8x³ - 6x( x² - y² + 4yz - 4z²)
=2x³ - 6xy² + 24z²x - 24xyz

thus,

P-Q=( 2x³ + 6xy² + 24z²x + 24xyz) - ( 2x³ + 6xy² + 24z²x - 24xyz)
　　　=48xyz

Homework Equations

I solved this problem like above, but the answer of the reference book is different from mine.
I can't this way to solve this problem. Please help me.

The answer of reference book is below

transpose　　 {x + y + 2z}³ - {　y＋2z - x }³=P,
{ 2z + x - y }³ + { x + y - 2z}³=Q

P={( x + y + 2z) - ( y + 2z - x)}{( x + y + 2z)² + ( x + y +2z )( y + 2z - x) + ( y + 2z - x)²}
=2x{ x² + 2x( y + 2z) + ( y + 2z)² + ( y + 2z)² - x² + (y + 2z)² - 2x( y + 2z) + x²}
=2x{3( y + 2z)² + x²}

Q={(2z + x - y) + ( x + y - 2z)}{(2z + x - y)² - ( 2z + x - y)( x + y - 2z) + (x + y - 2z)²}
=2x{(2z - y)² + 2x(2z - y) + x² - x² + (2z - y)² + (2z - y)² - 2x( 2z - y) + x²}
=2x{3(2z - y)² + x²}

thus,

P-Q=2x{3(y + 2z)² + x² - 3( 2z - y)²- x²}
=6x・8yz=48xyz
the answer is same to my result, but the way to solve this problem is different. I want to understand the solving way of reference book. Please help me.

 

[Mentallic]

The reference book is using the rule

$$a^3+b^3=(a+b)(a^2+ab+b^2)$$

and

$$a^3-b^3=(a-b)(a^2-ab+b^2)$$

 

[haruspex]

The reference book is using the rule

$$a^3+b^3=(a+b)(a^2+ab+b^2)$$

and

$$a^3-b^3=(a-b)(a^2-ab+b^2)$$

You mean

$$a^3+b^3=(a+b)(a^2-ab+b^2)$$

and

$$a^3-b^3=(a-b)(a^2+ab+b^2)$$

 

[Mentallic]

You mean

$$a^3+b^3=(a+b)(a^2-ab+b^2)$$

and

$$a^3-b^3=(a-b)(a^2+ab+b^2)$$

Yeah

 

[superduck]

I noticed that, but I could find my mistake

Yes, that formula is true. I found that the first comment has a little mistake. But, I could find my mistake by the comment. I'm grateful for two members, thank you very much.
I'm poor at English and also mathematics, so if you advise me when I ask for help, I'll be really grateful for that.