`(a+b)^3-(a-b)^3` can be factorized as:

Options

A. `2a(a^2+3b^2)`
B. `2a(3a^2+b^2)`
C. `2b(3a^2+b^2)`
D. `2a(a^2+3b^2)`

Answer

To factorize the expression \((a+b)^3 - (a-b)^3\), we can follow these steps:

### Step 1: Expand both cubes
We start by using the formula for the cube of a binomial, which states that:
\[ (x+y)^3 = x^3 + y^3 + 3xy(x+y) \]
Applying this to both \((a+b)^3\) and \((a-b)^3\):

1. For \((a+b)^3\):
\[ (a+b)^3 = a^3 + b^3 + 3ab(a+b) \]

2. For \((a-b)^3\):
\[ (a-b)^3 = a^3 - b^3 - 3ab(a-b) \]

### Step 2: Substitute the expansions into the expression
Now we substitute these expansions into the original expression:
\[ (a+b)^3 - (a-b)^3 = \left(a^3 + b^3 + 3ab(a+b)\right) - \left(a^3 - b^3 - 3ab(a-b)\right) \]

### Step 3: Simplify the expression
Distributing the negative sign in the second part:
\[ = a^3 + b^3 + 3ab(a+b) - a^3 + b^3 + 3ab(a-b) \]
Now, combine like terms:
\[ = (a^3 - a^3) + (b^3 + b^3) + 3ab(a+b) + 3ab(a-b) \]
This simplifies to:
\[ = 2b^3 + 3ab(a+b) + 3ab(a-b) \]

### Step 4: Combine the terms involving \(ab\)
Now we combine the terms involving \(ab\):
\[ 3ab(a+b) + 3ab(a-b) = 3ab(a+b + a-b) = 3ab(2a) = 6a^2b \]
So now we have:
\[ = 2b^3 + 6a^2b \]

### Step 5: Factor out the common terms
We can factor out \(2b\) from the expression:
\[ = 2b(b^2 + 3a^2) \]

### Final Result
Thus, the factorization of \((a+b)^3 - (a-b)^3\) is:
\[ \boxed{2b(b^2 + 3a^2)} \]