[math: 4x+1(x+4)2(x2+4)] \frac{4x+1}{(x+4)^2(x^2+4)} can be expressed in partial fractions as:

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Question restatement: We are asked to express the fraction (4x+1)/((x+4)^2 (x^2+4)) in partial fractions. The provided data shows an intended form: b. [Ax+4+B(x+4)2+Cx+Dx2+4] A/(x+4) + B/(x+4)^2 + (Cx+D)/(x^2+4).

First, recall the standard partial fraction decomposition approach for a rational function with a repeated linear factor and an irreducible quadratic: when the denominator has (x+4)^2 and (x^2+4), the appropriate decomposition is of the form A/(x+4) + B/(x+4)^2 + (Cx+D)/(x^2+4). The idea is to account for the double root at x = -4 with two distinct terms (one for the simple pole and one for the repeated pole), and to represent the irreducible quadratic factor x^2+4 with a linear numerator (Cx+D).

Now, analyze the structure of the proposed option (as written): it seems to attempt to present the same standard form, albeit with typographical and formatting issues. The intended components appear to be:
- A term over (x+4): A/(x+4)
- A term over (x+4)^2: B/(x+4)^2
- A term over the quadratic factor with a linear numerator: (Cx+D)/(x^2+4)
This aligns with the known correct structure for this denominator, where A, B, C, D are constants to be determined.

Why this form is the correct structural choice (conceptual):
- The denominator factors into (x+4)^2 and (x^2+4). For a repeated linear factor (x+4)^2, we need two terms to capture the residues at that factor: one with denominator (x+4) and one with (x+4)^2.
- The quadratic factor x^2+4 is irreducible over the reals, so its numerator must be of degree less than 2, i.e., a linear expression Cx+D.
- This setup ensures that, after combining the partial fractions over a common denominator, the numerator will be a polynomial of degree at most 3 (since the original numerator is degree 1 and the common denominator is degree 6), allowing us to solve for A, B, C, D by equating coefficients.

Regarding the provided answer options: the data shows an answer field that attempts to present the decomposition in the canonical form described above, but the options list is empty. This means there are no alternative decompositions to compare against. If one were to solve for A, B, C, D, one would substitute the decomposition back into (4x+1)/((x+4)^2 (x^2+4)), multiply through by the common denominator to obtain an identity in x, and then equate coefficients of like powers of x to solve for A, B, C, D. The specific numerical values would be determined by that system, but they are not provided here.

In summary, the structurally correct partial fraction template for the given denominator is A/(x+4) + B/(x+4)^2 + (Cx+D)/(x^2+4). The presented option mirrors this form, aside from formatting glitches, and the absence of additional options means there is nothing else to compare or evaluate beyond recognizing the standard decomposition pattern.

[math: 4x+1(x+4)2(x2+4)] \frac{4x+1}{(x+4)^2(x^2+4)} can be expressed in partial fractions as:

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