From the image, the question appears to be a ratio problem involving algebraic expressions:

The problem reads:

$\frac{2x+5}{(x+1)^4} = \frac{A}{(x+1)^3} + \frac{B}{(x+1)^4}$

We are tasked with solving for the values of $A$ and $B$.

Steps:

1. Combine the terms on the right-hand side by getting a common denominator.

   The common denominator of $(x+1)^3$ and $(x+1)^4$ is $(x+1)^4$. Therefore, we rewrite the right side: $\frac{A}{(x+1)^3} + \frac{B}{(x+1)^4} = \frac{A(x+1)}{(x+1)^4} + \frac{B}{(x+1)^4}$ $= \frac{A(x+1) + B}{(x+1)^4}$

2. Equating the numerators: Since both sides have the same denominator $(x+1)^4$, we can equate the numerators: $2x + 5 = A(x + 1) + B$

3. Expand the terms on the right-hand side: $2x + 5 = A(x) + A + B$ $2x + 5 = A x + (A + B)$

4. Match the coefficients:

   • The coefficient of $x$ on the left side is 2, and on the right side it is $A$, so: $A = 2$
   • The constant term on the left side is 5, and on the right side it is $A + B$, so: $A + B = 5$

5. Solve for $B$:

   • Since $A = 2$, substitute into $A + B = 5$: $2 + B = 5$ $B = 3$

Final Answer:

• $A = 2$
• $B = 3$

Would you like any further clarification or additional steps?

Here are some related questions you might find helpful:

1. How can you solve similar ratio problems with higher degree polynomials?
2. What is the process of factoring the denominator to find common denominators?
3. How do you check your solution in algebraic fraction problems?
4. What happens if we have more than two terms on the right-hand side of the equation?
5. How do you handle equations with different exponents of $x$ in the numerator and denominator?

Tip: When working with algebraic fractions, always look for a common denominator, as it allows you to combine or compare the terms more easily.