To solve the given problem, we need to follow these steps:

1. Apply the transformation $T_1$ to the polynomial $-3x^2 + 4x + 5$.
2. Use the result from step 1 as input for the transformation $T_2$.

Given: $T_1(ax^2 + bx + c) = \begin{bmatrix} a + b \\ b - c \end{bmatrix}$ $T_2 \left( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right) = \begin{bmatrix} 4x_1 & -8x_1 \\ 9x_2 & 5x_2 \end{bmatrix}$

First, we identify $a$, $b$, and $c$ in the polynomial $-3x^2 + 4x + 5$:

$a = -3, \quad b = 4, \quad c = 5$

Step 1: Apply $T_1$

Using the definition of $T_1$:

$T_1(-3x^2 + 4x + 5) = \begin{bmatrix} -3 + 4 \\ 4 - 5 \end{bmatrix} = \begin{bmatrix} 1 \\ -1 \end{bmatrix}$

Step 2: Apply $T_2$

Now, we take the result from $T_1$, which is $\begin{bmatrix} 1 \\ -1 \end{bmatrix}$, and apply $T_2$:

$T_2 \left( \begin{bmatrix} 1 \\ -1 \end{bmatrix} \right) = \begin{bmatrix} 4 \cdot 1 & -8 \cdot 1 \\ 9 \cdot (-1) & 5 \cdot (-1) \end{bmatrix} = \begin{bmatrix} 4 & -8 \\ -9 & -5 \end{bmatrix}$

Thus, the result of the linear transformation $(T_2 \circ T_1)(-3x^2 + 4x + 5)$ is:

$\begin{bmatrix} 4 & -8 \\ -9 & -5 \end{bmatrix}$

Would you like more details or have any questions about this solution?

Here are some additional related questions:

1. How do linear transformations affect the basis of a vector space?
2. Can you find the matrix representation of $T_1$ and $T_2$?
3. What properties do linear transformations preserve?
4. How do you compose multiple linear transformations?
5. How do linear transformations relate to systems of linear equations?
6. What is the kernel of a linear transformation?
7. How do you determine if a linear transformation is invertible?
8. What is the significance of eigenvalues and eigenvectors in linear transformations?

Tip: When working with linear transformations, always check the definitions and properties to ensure the correct application of each transformation.