**Problem:**
For which numbers

will the function

satisfy

for all

?

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**Attempt:** Correct up to here.
By using polynomial division I get the following result:

The trouble with this is that you are about to find that a = –d, so the denominator of the fraction in parentheses is zero.
By doing some hopefully correct algebra I get:

1.

So,

2.

So,

If

and

have opposite signs, their product is negative, and they can be any number.

If on the other hand, their signs are the same, their product is positive and:

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Am I making this more complicated than it is?

How can I better express my results regarding

, and

?