Factorise : `40(3x + 4y)^(2) + 14(3x + 4y) + 1`

Options

A. `(12x + 16y + 4)(30x + 40y + 10)`
B. `(12x + 16y + 1)(30x + 40y + 1)`
C. `(3x + 4y + 4)(3x + 4y + 10)`
D. `(16x + 12y + 4)(40x + 30y + 10)`

Answer

To factorize the expression \( 40(3x + 4y)^2 + 14(3x + 4y) + 1 \), we can follow these steps:

### Step 1: Substitute
Let \( T = 3x + 4y \). Then, we can rewrite the expression in terms of \( T \):
\[ 40T^2 + 14T + 1 \]

**Hint**: Substituting a variable can simplify the expression, making it easier to factor.

### Step 2: Factor the Quadratic
Now, we need to factor the quadratic expression \( 40T^2 + 14T + 1 \). We look for two numbers that multiply to \( 40 \times 1 = 40 \) and add up to \( 14 \). The numbers that satisfy this condition are \( 10 \) and \( 4 \).

**Hint**: When factoring quadratics, look for two numbers that multiply to the product of the leading coefficient and the constant term, and add to the middle coefficient.

### Step 3: Rewrite the Middle Term
We can rewrite the expression as:
\[ 40T^2 + 10T + 4T + 1 \]

**Hint**: Breaking down the middle term helps in grouping the expression for factoring.

### Step 4: Group the Terms
Now, we group the terms:
\[ (40T^2 + 10T) + (4T + 1) \]

**Hint**: Grouping terms allows us to factor by grouping.

### Step 5: Factor Each Group
Now, we factor out the common factors from each group:
\[ 10T(4T + 1) + 1(4T + 1) \]

**Hint**: Look for common factors in each group to simplify the expression.

### Step 6: Factor Out the Common Binomial
Now we can factor out the common binomial \( (4T + 1) \):
\[ (10T + 1)(4T + 1) \]

**Hint**: Factoring out common binomials is a key step in simplifying expressions.

### Step 7: Substitute Back
Now, we substitute back \( T = 3x + 4y \):
\[ (10(3x + 4y) + 1)(4(3x + 4y) + 1) \]

### Step 8: Simplify
Now, we simplify each factor:
1. \( 10(3x + 4y) + 1 = 30x + 40y + 1 \)
2. \( 4(3x + 4y) + 1 = 12x + 16y + 1 \)

Thus, the final factorized form of the expression is:
\[ (30x + 40y + 1)(12x + 16y + 1) \]

**Final Answer**:
\[ (30x + 40y + 1)(12x + 16y + 1) \]