How to Completely Factor the Expression 25x² – 36 – 25x + 6x + 6 – 25x + 6x + 6 – 5x + 6 – 5x + 6?

To factor the given expression completely, we start by simplifying it. The expression can be rewritten by combining like terms:

25x² - 25x + 6x - 25x + 6 + 6 - 5x + 6 - 5x + 6

After combining the x terms, we have:

25x² - 25x - 5x + 6 + 6 + 6 + 6

Now simplifying the constants:

25x² - 25x - 5x + 24

This results in:

25x² - 30x + 24

Next step is to factor the quadratic expression. We can look for two numbers whose product is (25 * 24) = 600 and whose sum is (-30). The numbers are -24 and -6. We can rewrite the middle term:

25x² - 24x - 6x + 24

Now factoring by grouping, we group the first two and the last two terms:

(25x² - 24x) + (-6x + 24)

Factoring out the common terms from each group, we get:

x(25x - 24) - 6(25x - 4)

Thus, the expression can be factored as:

(25x - 24)(x - 6)

Therefore, the completely factored form of the expression is:

(25x - 24)(x - 6)