The factors of 3(x + 2)² + 5(x + 2) + 2 are :

Question

Accepted Answer

To determine why Option 2 is the correct factorization of the expression $3(x + 2)^2 + 5(x + 2) + 2$, let's go through the process of expanding and factoring the expression.

### Step-by-Step Explanation:

1. **Expand the Expression:**
   - Start with the expression: $3(x + 2)^2 + 5(x + 2) + 2$.
   - First, expand $(x + 2)^2$: 
     $$
     (x + 2)^2 = x^2 + 4x + 4
     $$
   - Substitute back into the expression:
     $$
     3(x^2 + 4x + 4) + 5(x + 2) + 2
     $$
   - Distribute the 3:
     $$
     3x^2 + 12x + 12 + 5x + 10 + 2
     $$
   - Combine like terms:
     $$
     3x^2 + 17x + 24
     $$

2. **Factor the Quadratic Expression:**
   - We need to factor $3x^2 + 17x + 24$.
   - Look for two numbers that multiply to $3 \times 24 = 72$ and add to 17. These numbers are 8 and 9.
   - Rewrite the middle term using 8 and 9:
     $$
     3x^2 + 8x + 9x + 24
     $$
   - Factor by grouping:
     $$
     (3x^2 + 8x) + (9x + 24)
     $$
   - Factor out the greatest common factor from each group:
     $$
     x(3x + 8) + 3(3x + 8)
     $$
   - Notice the common factor $(3x + 8)$:
     $$
     (3x + 8)(x + 3)
     $$

### Justification for Option 2:

- The factorization $(3x + 8)(x + 3)$ matches Option 2 exactly. This confirms that Option 2 is the correct factorization of the given expression.

### Explanation of Why Other Options Are Incorrect:

1. **Option 1: (3x + 5)(x + 2)**
   - Expanding this gives $3x^2 + 6x + 5x + 10 = 3x^2 + 11x + 10$, which does not match the original expression $3x^2 + 17x + 24$.

2. **Option 3: (3x + 8)(x + 5)**
   - Expanding this gives $3x^2 + 15x + 8x + 40 = 3x^2 + 23x + 40$, which does not match the original expression.

3. **Option 4: (3x + 1)(x + 2)**
   - Expanding this gives $3x^2 + 6x + x + 2 = 3x^2 + 7x + 2$, which does not match the original expression.

Thus, Option 2 is the only factorization that correctly represents the expanded form of the given expression, confirming it as the correct answer.