Completely Factoring the Expression
To factor the expression 25x² + 36 + 25x + 6x + 6 + 25x + 6x + 6 + 5x + 65x + 6 + 5x + 65x + 6, we first need to simplify it.
Step 1: Combine Like Terms
Combining all the like terms, we have:
- For the x² term: 25x²
- For the x terms: 25x + 25x + 6x + 6x + 5x + 65x + 5x + 65x = 25x + 10x + 130x = 130x
- For the constant terms: 36 + 6 + 6 + 6 + 6 = 60
So, our expression becomes:
25x² + 130x + 60
Step 2: Factoring the Quadratic Expression
Next, we will factor the simplified quadratic:
We need two numbers that multiply to (25 * 60) = 1500 and add to 130. These numbers are 120 and 10.
Now, we can rewrite the middle term:
25x² + 120x + 10x + 60
Step 3: Grouping
We will group the terms:
(25x² + 120x) + (10x + 60)
Factoring out the common factors:
5x(5x + 24) + 10(1x + 6)
Step 4: Final Factoring
Now we can factor out the common binomial:
(5x + 10)(5x + 24)
Final Result
The completely factored form of the expression 25x² + 36 + 25x + 6x + 6 + 25x + 6x + 6 + 5x + 65x + 6 + 5x + 65x + 6 is:
(5x + 10)(5x + 24)