Factorise : `27 + 125 a ^(3) + 135 a + 225 a ^(2)`

Options

A. `(3 + 5a) (3 + 5a) (3 - 5a)`
B. `(3 - 5a) (3 - 5a) (3 + 5a)`
C. `(3 + 5a) (3 + 5a) (3 + 5a)`
D. `(3 -5a) (3 + 5a) (3 + 5a)`

Answer

To factorize the expression \( 27 + 125a^3 + 135a + 225a^2 \), we can follow these steps:

### Step 1: Rearrange the expression
We can rearrange the expression in a more standard form:
\[ 125a^3 + 225a^2 + 135a + 27 \]

### Step 2: Identify the cubes
Notice that \( 27 \) can be written as \( 3^3 \) and \( 125a^3 \) can be written as \( (5a)^3 \). So we have:
\[ 27 = 3^3 \]
\[ 125a^3 = (5a)^3 \]

### Step 3: Rewrite the expression
Now, we can rewrite the expression as:
\[ (5a)^3 + 3^3 + 135a + 225a^2 \]

### Step 4: Factor out common terms
Next, we can factor out the common terms from the last two terms:
\[ 135a + 225a^2 = 45a(3 + 5a) \]

### Step 5: Combine the expression
Now, we can rewrite the entire expression:
\[ (5a)^3 + 3^3 + 45a(3 + 5a) \]

### Step 6: Recognize the sum of cubes
The expression \( (5a)^3 + 3^3 \) can be factored using the sum of cubes formula:
\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]
where \( x = 5a \) and \( y = 3 \).

### Step 7: Apply the sum of cubes formula
Applying the formula:
\[ (5a + 3)((5a)^2 - (5a)(3) + 3^2) \]
Calculating this gives:
\[ (5a + 3)(25a^2 - 15a + 9) \]

### Step 8: Combine with the factored part
Now, we need to combine this with the factored part \( 45a(3 + 5a) \):
Thus, the complete factorization is:
\[ (5a + 3)(25a^2 - 15a + 9) + 45a(3 + 5a) \]

### Final Factorization
The final factorization of the expression \( 27 + 125a^3 + 135a + 225a^2 \) is:
\[ (3 + 5a)^3 \]

---