The problem asks to determine the $2 \times 2$ matrix $X$, given that:

$A = \begin{bmatrix} 4 & 2 \\ 0 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & -4 \\ 3 & 5 \end{bmatrix},$ and the equation: $2A + X = 2B.$

Step-by-step solution:

1. Rewrite the equation: From $2A + X = 2B$, solve for $X$: $X = 2B - 2A.$

2. Calculate $2A$: Multiply each element of $A$ by 2: $2A = 2 \cdot \begin{bmatrix} 4 & 2 \\ 0 & 3 \end{bmatrix} = \begin{bmatrix} 8 & 4 \\ 0 & 6 \end{bmatrix}.$

3. Calculate $2B$: Multiply each element of $B$ by 2: $2B = 2 \cdot \begin{bmatrix} 1 & -4 \\ 3 & 5 \end{bmatrix} = \begin{bmatrix} 2 & -8 \\ 6 & 10 \end{bmatrix}.$

4. Subtract $2A$ from $2B$: Compute $X = 2B - 2A$: $X = \begin{bmatrix} 2 & -8 \\ 6 & 10 \end{bmatrix} - \begin{bmatrix} 8 & 4 \\ 0 & 6 \end{bmatrix} = \begin{bmatrix} 2 - 8 & -8 - 4 \\ 6 - 0 & 10 - 6 \end{bmatrix}.$

5. Simplify the result: $X = \begin{bmatrix} -6 & -12 \\ 6 & 4 \end{bmatrix}.$

Final Answer:

$X = \begin{bmatrix} -6 & -12 \\ 6 & 4 \end{bmatrix}.$

Do you have any questions, or would you like further details?

Here are 5 related questions for further exploration:

1. What is the determinant of the resulting matrix $X$?
2. How can matrix addition and scalar multiplication be visualized geometrically?
3. How would $X$ change if $2A$ were replaced with $3A$?
4. Can you verify $X$ by substituting it back into the original equation?
5. What is the trace of the matrix $X$?

Tip: When performing matrix operations, always verify dimensions to ensure compatibility!