To factor the expression completely, we first need to simplify it. We combine like terms:
- For the x² terms: 4x² + 32x² = 36x²
- For the x terms: 25x + 4x + 6 + 4x + 6x + 2x + 62x = 109x
- Constant terms: 1 + 1 + 1 + 2 = 5
Now our expression is:
36x² + 109x + 5
Next, we look for two numbers that multiply to (36 * 5 = 180) and add to 109. The numbers 108 and 1 fit this requirement. We can now rewrite the middle term:
36x² + 108x + x + 5
Next, we group the terms:
(36x² + 108x) + (x + 5)
From the first group, we can factor out 36x:
36x(x + 3) + 1(x + 5)
Now, we can factor out the common binomial factor:
(6x + 1)(6x + 5) = 0
Thus, the complete factorization of the original expression is:
(6x + 1)(6x + 5)