To determine the constants $A$, $B$, $C$, and $D$, we need to express the given rational function on the left side in terms of the partial fractions on the right side.

Given: $\frac{3x^3 + x^2 - 7x - 5}{(x^2 + 2x + 3)^2} = \frac{Ax + B}{x^2 + 2x + 3} + \frac{Cx + D}{(x^2 + 2x + 3)^2}$

1. Combine terms on the right side: Rewrite the right side over a common denominator: $\frac{Ax + B}{x^2 + 2x + 3} + \frac{Cx + D}{(x^2 + 2x + 3)^2} = \frac{(Ax + B)(x^2 + 2x + 3) + (Cx + D)}{(x^2 + 2x + 3)^2}$

2. Expand the numerator: Expand $(Ax + B)(x^2 + 2x + 3)$ to get: $(Ax + B)(x^2 + 2x + 3) = Ax^3 + 2Ax^2 + 3Ax + Bx^2 + 2Bx + 3B$ Simplify the terms: $= Ax^3 + (2A + B)x^2 + (3A + 2B)x + 3B$

3. Combine terms in the numerator: Now the numerator of the right side is: $Ax^3 + (2A + B)x^2 + (3A + 2B)x + 3B + Cx + D$ Simplify further: $= Ax^3 + (2A + B)x^2 + (3A + 2B + C)x + (3B + D)$

4. Set up a system of equations: Since this must equal the numerator on the left side, we equate the coefficients of each power of $x$ in the numerators:

   • For $x^3$: $A = 3$
   • For $x^2$: $2A + B = 1$
   • For $x$: $3A + 2B + C = -7$
   • For the constant term: $3B + D = -5$

5. Solve the system of equations:

   • From $A = 3$, substitute $A = 3$ into the other equations:
     • $2(3) + B = 1 \Rightarrow B = -5$
     • $3(3) + 2(-5) + C = -7 \Rightarrow C = -4$
     • $3(-5) + D = -5 \Rightarrow D = 10$

Thus, the values of the constants are: $A = 3, \quad B = -5, \quad C = -4, \quad D = 10$

Would you like to delve into the solution details or explore similar examples?

1. How do we apply partial fraction decomposition for higher-order denominators?
2. What techniques simplify solving systems of equations in partial fractions?
3. Why is equating coefficients a reliable method in polynomial equations?
4. Can we generalize this method to other rational functions with repeated factors?
5. How does this technique connect to integration of rational functions?

Tip: When using partial fraction decomposition, always check if the degrees align properly for valid decomposition.