Final answer:
The given expression can be simplified by expanding and combining like terms in the numerator and then simplifying with the denominator to obtain 64 / (x + 3)^3.
Step-by-step explanation:
To simplify the expression [((x+3)^2)(2x+6) - (x^2 + 6x - 23)(2)(x+3)] / (x+3)^4, we start by expanding both the numerator and the part of the denominator that isn't immediately obvious. Let's work on the numerator first.
Expand (x+3)^2 to get x^2 + 6x + 9, and then multiply by (2x+6) to get 2x^3 + 12x^2 + 18x + 6x^2 + 36x + 54. Combining like terms, we get 2x^3 + 18x^2 + 54x + 54.
Next, we expand (x^2 + 6x - 23)(2)(x+3), which is 2(x^2 + 6x - 23)(x + 3). This equals to 2(x^3 + 3x^2 + 6x^2 + 18x - 23x - 69). Combining the terms results in 2(x^3 + 9x^2 - 5x - 69). Doubling each term, we get 2x^3 + 18x^2 - 10x - 138.
Now, we subtract the second expanded term from the first: (2x^3 + 18x^2 + 54x + 54) - (2x^3 + 18x^2 - 10x - 138). We get 64x + 192 in the numerator after combining like terms.
To simplify this numerator with the denominator, we note that the numerator has a common factor of (x + 3). We can write 64x + 192 as 64(x + 3). Then, this can be simplified with the (x+3)^4 in the denominator. Since (x + 3)^4 has a higher power than (x + 3), we end up with 64 / (x + 3)^3.