Final Answer:
The quotient t + 3/t + 4 ÷ (t^2 + 7t + 12) is 1/(t + 4).
Step-by-step explanation:
To find the quotient, we need to divide t + 3/t + 4 by (t^2 + 7t + 12). Here's how we can do it:
Step 1: Factor the denominator:
(t^2 + 7t + 12) can be factored as (t + 4)(t + 3).
Step 2: Rewrite the division as multiplication by the inverse:
Division is the same as multiplication by the inverse. Therefore, we can rewrite the expression as:
(t + 3)/(t + 4) * 1/((t + 4)(t + 3))
Step 3: Simplify the expression:
(t + 3)/(t + 4) * 1/((t + 4)(t + 3)) = (t + 3) * 1 / ((t + 4)(t + 3))
Step 4: Cancel out common factors:
We can cancel out the (t + 3) term in the numerator and denominator:
(t + 3) * 1 / ((t + 4)(t + 3)) = (t + 3) * 1 / (t + 4)(t + 3) = 1 / (t + 4)
Therefore, the quotient of t + 3/t + 4 ÷ (t^2 + 7t + 12) is 1/(t + 4).