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The image you uploaded contains a set of mathematical problems labeled from 1 to 4, and an additional problem with an "Answer" line.

I'll work through each of the problems one by one and provide the solutions.

### 1. \(\text{Evaluate } \frac{4x^2 + 3x - 7}{x^2 - 5x + 6} \text{ for } x = -1\)
   Substitute \(x = -1\) into the expression:
   \[
   \text{Numerator: } 4(-1)^2 + 3(-1) - 7 = 4(1) - 3 - 7 = 4 - 3 - 7 = -6
   \]
   \[
   \text{Denominator: } (-1)^2 - 5(-1) + 6 = 1 + 5 + 6 = 12
   \]
   \[
   \text{Value of the function: } \frac{-6}{12} = -\frac{1}{2}
   \]
   **Answer:** \(-\frac{1}{2}\)

### 2. \(\text{Let } y = \sqrt{x^2 + 2x + 1}, \text{ evaluate } y \text{ at } x = -3\)
   Substitute \(x = -3\) into the expression:
   \[
   y = \sqrt{(-3)^2 + 2(-3) + 1} = \sqrt{9 - 6 + 1} = \sqrt{4} = 2
   \]
   **Answer:** \(2\)

### 3. \(f(x) = \frac{x^2 + 3}{x - 4}, \text{ evaluate } f(-3)\)
   Substitute \(x = -3\) into the expression:
   \[
   \text{Numerator: } (-3)^2 + 3 = 9 + 3 = 12
   \]
   \[
   \text{Denominator: } -3 - 4 = -7
   \]
   \[
   \text{Value of the function: } f(-3) = \frac{12}{-7} = -\frac{12}{7}
   \]
   **Answer:** \(-\frac{12}{7}\)

### 4. \(\text{If } y = 2x^2 + 3x - 5, \text{ find } f(2a - 3)\)
   To find \(f(2a - 3)\), substitute \(x = 2a - 3\) into the expression for \(y\):
   \[
   y = 2(2a - 3)^2 + 3(2a - 3) - 5
   \]
   Expanding \( (2a - 3)^2 \):
   \[
   (2a - 3)^2 = 4a^2 - 12a + 9
   \]
   So:
   \[
   y = 2(4a^2 - 12a + 9) + 3(2a - 3) - 5 = 8a^2 - 24a + 18 + 6a - 9 - 5 = 8a^2 - 18a + 4
   \]
   **Answer:** \(8a^2 - 18a + 4\)

Would you like further details or explanations on any of the steps?  

Here are some related questions you might want to explore:
1. What happens if the denominator in a function evaluation is zero?
2. How do you factor quadratic expressions?
3. What are the steps to simplify square roots involving expressions?
4. How do you expand and simplify binomials like \((2a - 3)^2\)?
5. What are the steps to solve quadratic equations by completing the square?
6. How do you evaluate functions for different variable inputs?
7. What are the rules for simplifying rational expressions?
8. How can you check if a function is undefined at certain values of \(x\)?

**Tip:** When evaluating functions, always ensure the denominator is non-zero to avoid undefined values.

Explore how to evaluate mathematical functions and solve quadratic equations with detailed step-by-step solutions. This problem involves algebraic concepts such as function evaluation and the quadratic formula, suitable for advanced high school students.

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