Factorise:
`49 (2x + 3y) ^(2) - 70 ( 4x ^(2) - 9 y ^(2)) + 25 ( 2 x - 3y ) ^(2)`

Options

A. `4 ( x - 9y) ^(2)`
B. `9 (x + 4y) ^(2)`
C. `16 (x + 9 y ) ^(2)`
D. `16 (x - 9y) ^(2)`

Answer

To factorize the expression \( 49 (2x + 3y)^2 - 70 (4x^2 - 9y^2) + 25 (2x - 3y)^2 \), we will follow these steps:

### Step 1: Rewrite the expression
We start by rewriting the expression clearly:
\[ 49 (2x + 3y)^2 - 70 (4x^2 - 9y^2) + 25 (2x - 3y)^2 \]

### Step 2: Recognize and apply the difference of squares
Notice that \( 4x^2 - 9y^2 \) can be factored as a difference of squares:
\[ 4x^2 - 9y^2 = (2x)^2 - (3y)^2 = (2x + 3y)(2x - 3y) \]
Thus, we can rewrite the expression:
\[ 49 (2x + 3y)^2 - 70 \cdot (2x + 3y)(2x - 3y) + 25 (2x - 3y)^2 \]

### Step 3: Substitute variables for simplification
Let \( a = 2x + 3y \) and \( b = 2x - 3y \). The expression now becomes:
\[ 49a^2 - 70ab + 25b^2 \]

### Step 4: Recognize the quadratic form
This expression resembles a quadratic in terms of \( a \) and \( b \):
\[ 49a^2 - 70ab + 25b^2 \]
We can factor this quadratic expression using the formula \( A^2 - 2AB + B^2 = (A - B)^2 \):
\[ (7a - 5b)^2 \]

### Step 5: Substitute back the original variables
Now we substitute back \( a \) and \( b \):
\[ (7(2x + 3y) - 5(2x - 3y))^2 \]

### Step 6: Simplify the expression
Now simplify the expression inside the parentheses:
\[ 7(2x + 3y) - 5(2x - 3y) = 14x + 21y - 10x + 15y = (14x - 10x) + (21y + 15y) = 4x + 36y \]

### Final Step: Write the final factorized form
Thus, the factorized form of the original expression is:
\[ (4x + 36y)^2 \]

### Summary of the Factorization
The final answer is:
\[ (4x + 36y)^2 \]