Sure, let's start.
We have two functions, f(x) and g(x), which are given by the following equations:
f(x) = x^2 + 3x - 28
g(x) = x + 7
We're asked to find the rule for the quotient of these functions, (f)/(g). This is equivalent to dividing f(x) by g(x):
So (f/g)(x) = (x^2 + 3x - 28) / (x + 7)
You can see in the denominator, (x + 7), of the resulting equation, that it's a factor of the polynomial in the numerator, (x^2 + 3x - 28).
We can factorize the numerator as:
(x - 4)(x + 7)
When we divide the numerator by the denominator, we only keep the unfactored portion of the numerator since both the numerator and the denominator share x + 7 as a factor.
So when you divide the whole expression by (x + 7) you will have:
(f/g)(x) = (x - 4)
Hence, our answer is x - 4. It means the resultant function after diving f(x) by g(x) will be another function (f/g)(x) = x - 4 which is a straight line passing through point (4, 0) on the x-axis and whose slope is equivalent to 1.