least common denominator of (x^5/x^2 6x 9) and (5/x^2 5x 6)

To find the least common denominator (LCD) of two fractions, we first need to factor the denominators of both fractions completely. Then, we find the common factors and select the highest power of each factor. Finally, we multiply all these factors together to obtain the LCD.

Let's factor the denominators:

Denominator 1: x^2 + 6x + 9
To factor this quadratic trinomial, we need to find two numbers that add up to 6 (the coefficient of x) and multiply to give us 9. In this case, these numbers are 3 and 3. So, the factored form is (x + 3)(x + 3) or (x + 3)^2.

Denominator 2: x^2 + 5x + 6
To factor this quadratic trinomial, we need to find two numbers that add up to 5 (the coefficient of x) and multiply to give us 6. In this case, these numbers are 2 and 3. So, the factored form is (x + 2)(x + 3).

Now, we can identify the common factors and select the highest power for each:

(x + 2), (x + 3)^2.

To find the LCD, we multiply the common factors:

(x + 2)(x + 3)^2 = (x + 2)(x^2 + 6x + 9) = x^3 + 8x^2 + 27x + 18.

Therefore, the least common denominator (LCD) of the given fractions is x^3 + 8x^2 + 27x + 18.