We are given a matrix equation and need to solve for the variables $a$ and $b$. Here's a breakdown of the matrix equation:

$\begin{pmatrix} 2a+3 & 8 \\ -3 & 4 \end{pmatrix} + \begin{pmatrix} -2 & b+4 \\ 2 & -3 \end{pmatrix} = \begin{pmatrix} 3 & 15 \\ 5 & 1 \end{pmatrix}$

Step-by-Step Approach:

1. Matrix Addition: Add the corresponding elements of the two matrices on the left-hand side.

   $\begin{pmatrix} (2a+3) + (-2) & 8 + (b+4) \\ -3 + 2 & 4 + (-3) \end{pmatrix} = \begin{pmatrix} 3 & 15 \\ 5 & 1 \end{pmatrix}$

2. Simplify Each Element:

   • For the element in position (1,1): $(2a+3) - 2 = 3$
   • For the element in position (1,2): $8 + (b+4) = 15$
   • For the element in position (2,1): $-3 + 2 = 5$
   • For the element in position (2,2): $4 - 3 = 1$

Solve for $a$ and $b$:

• From the first equation (1,1): $(2a + 3) - 2 = 3 \quad \Rightarrow \quad 2a + 1 = 3 \quad \Rightarrow \quad 2a = 2 \quad \Rightarrow \quad a = 1$

• From the second equation (1,2): $8 + (b + 4) = 15 \quad \Rightarrow \quad b + 12 = 15 \quad \Rightarrow \quad b = 3$

Thus, $a = 1$ and $b = 3$.

Part (b) of the Question:

We need to determine the values of the following:

1. $2a + 3$
2. $b + 4$

• Since $a = 1$, $2a + 3 = 2(1) + 3 = 2 + 3 = 5$.
• Since $b = 3$, $b + 4 = 3 + 4 = 7$.

Final Answers:

• $a = 1$
• $b = 3$
• $2a + 3 = 5$
• $b + 4 = 7$

Let me know if you want further explanation or details on any of the steps!

Here are five related questions for you to explore:

1. How do you add two matrices of different dimensions?
2. Can matrix equations have multiple solutions for $a$ and $b$?
3. How would the solution change if the given matrices were 3x3 instead of 2x2?
4. What is the geometric interpretation of matrix addition?
5. How do scalar multiplication and matrix addition differ in terms of operations?

Tip: When adding matrices, always ensure that the matrices have the same dimensions!