Final answer:
To solve the expression x² + 7x + 12/x² - 9 ÷ x² + 10x + 24/x² - 3x - 54 completely and state any restrictions on the variable, simplify the expressions first and then solve for x by setting the numerator equal to zero.
Step-by-step explanation:
To solve the expression x² + 7x + 12/x² - 9 ÷ x² + 10x + 24/x² - 3x - 54 completely, we need to simplify the expressions first.
Let's simplify the denominators:
x² - 9 can be factored as (x + 3)(x - 3)
x² + 10x + 24 can be factored as (x + 4)(x + 6)
x² - 3x - 54 can be factored as (x + 6)(x - 9)
Now that we have simplified the denominators, we can rewrite the expression as:
(x² + 7x + 12) / [(x + 3)(x - 3)] ÷ (x² + 10x + 24) / [(x + 4)(x + 6)] ÷ (x² - 3x - 54) / [(x + 6)(x - 9)]
Next, we can simplify further by multiplying the numerators and denominators:
(x² + 7x + 12)(x + 4)(x + 6) / [(x + 3)(x - 3)](x² + 10x + 24)(x + 6)(x - 9)
Now, we can add like terms and simplify the expression if possible.
To solve completely, we can solve for x by setting the numerator equal to zero and using algebraic techniques like factoring, quadratic formula, or completing the square.