To factor the expression completely, we first need to simplify it by combining like terms. Let’s break it down step by step.
The original expression is:
3x² + 5x + 1 + 3x + 1 + 3x + 5 + x + 1 + 3x + 5 + x + 1.
Now, let’s combine the terms:
- Combine the x² terms: 3x² (only one term).
- Combine the x terms: 5x + 3x + 3x + x + 3x = 15x.
- Combine the constant terms: 1 + 1 + 5 + 1 + 5 + 1 = 14.
So we can rewrite the expression as:
3x² + 15x + 14.
Now we will factor this quadratic expression. We look for two numbers that multiply to 3 * 14 = 42 and add to 15. The numbers that work are 6 and 7.
Now we can rewrite the middle term:
3x² + 6x + 7x + 14.
Next, we can group the terms:
- (3x² + 6x) + (7x + 14).
Factor out the common factors in each group:
- 3x(x + 2) + 7(x + 2).
Notice that (x + 2) is a common factor. Factor it out:
(x + 2)(3x + 7).
Since both factors (x + 2) and (3x + 7) don’t have any further common factors, we have the completely factored form:
(x + 2)(3x + 7).
Finally, check if the expression can be factored further:
Both factors are not factorable over the integers, meaning they are prime in this context.
Thus, the completely factored form of the original expression is:
(x + 2)(3x + 7).