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The division algorithm helps us break down more complex rational expressions. In simple terms, it states that given two polynomials, we can express one as the quotient times the divisor plus a remainder. This is very much similar to regular number division, but here, we work with polynomials. To solve our example problem, we start by dividing the numerator polynomial \(x^4 - 5x^2 + x - 4\) by the denominator polynomial \(x^2 + 4x + 4\). The result of this division gives us a quotient, which is another polynomial, and a remainder, which forms the basis of our proper rational expression.

A rational expression is proper when the degree (highest power) of the numerator is less than the degree of the denominator. If it's not, we need to perform polynomial long division to convert the improper rational expression into a proper one. After our division in the example, we get \(x^2 - 4x + 11\) as the polynomial and \(\frac{-43x + 40}{x^2 + 4x + 4}\) as the proper rational expression. This step is crucial as it ensures we can later apply partial fraction decomposition effectively.

Polynomial long division is the process we use to divide two polynomials. It’s similar to long division with numbers but adjusted for polynomials. Follow these steps to perform it:
- Divide the leading term of the numerator by the leading term of the denominator. This gives the first term of the quotient.
- Multiply the entire denominator by this term and subtract from the numerator.
- The result is the new numerator. Repeat the above steps with the new numerator until the degree of the remaining numerator is less than that of the denominator.
In our example, dividing \(x^4 - 5x^2 + x - 4\) by \(x^2 + 4x + 4\), we get the quotient \(x^2 - 4x + 11\) and the remainder \(-43x + 40\).

Partial fraction decomposition is the process of breaking down a proper rational expression into simpler fractions that are easier to integrate or work with. Once we have a proper rational expression, we set it up into a desirable form for decomposition. In our example, we decompose \(\frac{-43x + 40}{(x+2)^2}\) into simpler fractions. Assume:
\(\frac{-43x + 40}{(x+2)^2} = \frac{A}{x+2} + \frac{B}{(x+2)^2}\).
To find \(A\) and \(B\), we clear the fraction by multiplying through by \((x+2)^2\), equate coefficients of like terms to solve for \(A\) and \(B\). Here, \(A = -43\) and \(B = 126\), thus the decomposition is: \(\frac{-43x + 40}{(x+2)^2} = \frac{-43}{x+2} + \frac{126}{(x+2)^2}\). Finally, combining the polynomial and partial fractions, we get the final expression.