Let's work through the problems step by step.

Part A: Substitute the values of $x$ into the given functions to complete the table.

1. For $f(x) = x^2 + 6x + 5$:

   • $f(-2) = (-2)^2 + 6(-2) + 5 = 4 - 12 + 5 = -3$
   • $f(0) = (0)^2 + 6(0) + 5 = 5$
   • $f(1) = (1)^2 + 6(1) + 5 = 1 + 6 + 5 = 12$

   Result for $f(x)$:

   • $f(-2) = -3$
   • $f(0) = 5$
   • $f(1) = 12$

2. For $g(x) = (x + 3)^2 - 4$:

   • $g(-2) = ((-2) + 3)^2 - 4 = (1)^2 - 4 = 1 - 4 = -3$
   • $g(0) = (0 + 3)^2 - 4 = (3)^2 - 4 = 9 - 4 = 5$
   • $g(1) = (1 + 3)^2 - 4 = (4)^2 - 4 = 16 - 4 = 12$

   Result for $g(x)$:

   • $g(-2) = -3$
   • $g(0) = 5$
   • $g(1) = 12$

Part B: Transform $y = x^2 + 4x + 3$ to vertex form and write the characteristics.

1. Convert to Vertex Form: $y = a(x - h)^2 + k$

   • Given: $y = x^2 + 4x + 3$
   • Step 1: Complete the square:
     • $y = (x^2 + 4x) + 3$
     • Take half the coefficient of $x$, square it, and add/subtract: $(\frac{4}{2})^2 = 4$.
     • $y = (x^2 + 4x + 4 - 4) + 3 = ((x + 2)^2 - 4) + 3 = (x + 2)^2 - 1$

   Vertex Form: $y = (x + 2)^2 - 1$

2. Characteristics:

   • $a = 1$ (coefficient of $(x + 2)^2$)
   • $h = -2$ (opposite of $+2$ in the equation)
   • $k = -1$ (constant term in the vertex form)
   • Vertex: $(-2, -1)$
   • Line of symmetry: $x = -2$ (same as $h$)

Final Answers:

1. Table of Values:

   • $f(x)$:
     • $f(-2) = -3$, $f(0) = 5$, $f(1) = 12$
   • $g(x)$:
     • $g(-2) = -3$, $g(0) = 5$, $g(1) = 12$

2. Vertex Form and Characteristics:

   • Vertex Form: $y = (x + 2)^2 - 1$
   • $a = 1$, $h = -2$, $k = -1$
   • Vertex: $(-2, -1)$
   • Line of symmetry: $x = -2$

Would you like me to explain any step further or help with graphing these functions?

Related Questions:

1. How do you determine the vertex of a quadratic function using the formula $x = -\frac{b}{2a}$?
2. Why is completing the square useful for converting to vertex form?
3. What does the value of $a$ tell us about the graph of the quadratic function?
4. How does the line of symmetry relate to the vertex of a quadratic?
5. Can a quadratic function have multiple lines of symmetry? Why or why not?

Tip: When working with quadratic equations, always check whether factoring or completing the square is the easiest method for your problem!