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The problem is asking to find \( X \) such that:

\[
A^2 + 5A + BX = 0
\]

where the matrices \( A \) and \( B \) are given as:

\[
A = \begin{pmatrix} 2 & 5 \\ 4 & -3 \end{pmatrix}, \quad B = \begin{pmatrix} 2 & -1 \\ 4 & -2 \end{pmatrix}
\]

### Step-by-step Solution:

1. **Calculate \( A^2 \):**
   We need to square matrix \( A \).

   \[
   A^2 = A \times A = \begin{pmatrix} 2 & 5 \\ 4 & -3 \end{pmatrix} \times \begin{pmatrix} 2 & 5 \\ 4 & -3 \end{pmatrix}
   \]

   Let me calculate that.

2. **Multiply matrix \( A \) by 5:**
   Multiply each element of \( A \) by 5 to get \( 5A \).

3. **Set up the equation:**
   After computing \( A^2 \), \( 5A \), and inserting them into the equation \( A^2 + 5A + BX = 0 \), we will solve for \( X \).

Let me do the calculations.The calculations are as follows:

1. \( A^2 = \begin{pmatrix} 24 & -5 \\ -4 & 29 \end{pmatrix} \)
2. \( 5A = \begin{pmatrix} 10 & 25 \\ 20 & -15 \end{pmatrix} \)

Now, we substitute these into the equation \( A^2 + 5A + BX = 0 \):

\[
\begin{pmatrix} 24 & -5 \\ -4 & 29 \end{pmatrix} + \begin{pmatrix} 10 & 25 \\ 20 & -15 \end{pmatrix} + BX = 0
\]

Simplifying the left-hand side gives:

\[
\begin{pmatrix} 34 & 20 \\ 16 & 14 \end{pmatrix} + BX = 0
\]

This means:

\[
BX = \begin{pmatrix} -34 & -20 \\ -16 & -14 \end{pmatrix}
\]

Now, we solve for \( X \) by multiplying both sides by the inverse of matrix \( B \). Let me calculate that.It appears that matrix \( B \) is singular, meaning it does not have an inverse. Since \( BX = \begin{pmatrix} -34 & -20 \\ -16 & -14 \end{pmatrix} \) cannot be solved using matrix inversion, we will have to explore other methods such as solving the system by reducing it to simpler equations.

Let me proceed by attempting to solve this equation directly.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can walk you through manually solving the system \( BX = \begin{pmatrix} -34 & -20 \\ -16 & -14 \end{pmatrix} \) by using other methods such as row reduction or substitution.

Do you have any specific questions or would you like further assistance? Let me know if you'd like to continue with the solution manually.

Here are 5 related questions to deepen understanding:
1. What is a singular matrix, and why does it not have an inverse?
2. How can matrix equations be solved when a matrix is singular?
3. What is the least squares method, and how is it used in matrix equations?
4. Can matrix equations have multiple or no solutions, and under what conditions?
5. How does row reduction work in solving systems of linear equations?

**Tip:** When encountering a singular matrix, consider alternative methods like row reduction or pseudo-inverses for solving the system.

If A = [[2, 5], [4, -3]] and B = [[2, -1], [4, -2]], find X such that A^2 + 5A + BX = 0.

Learn how to solve the matrix equation A^2 + 5A + BX = 0 with matrices A and B. This problem involves matrix multiplication and matrix equation solving techniques, suitable for undergraduate students studying linear algebra.

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