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Let's complete the square for both equations.

### Part (a): \( y = x^2 + 10x + 12 \)

1. **Group the \(x\)-terms**:
   \[
   y = (x^2 + 10x) + 12
   \]

2. **Complete the square**:
   To complete the square, we take half of the coefficient of \(x\), which is \(10\), divide by 2 to get \(5\), and square it:
   \[
   \left(\frac{10}{2}\right)^2 = 25
   \]
   Add and subtract \(25\) inside the parentheses:
   \[
   y = (x^2 + 10x + 25 - 25) + 12
   \]
   \[
   y = ((x + 5)^2 - 25) + 12
   \]

3. **Simplify**:
   \[
   y = (x + 5)^2 - 13
   \]

Thus, the equation in completed square form is:
\[
y = (x + 5)^2 - 13
\]

---

### Part (b): \( y = -3x^2 - 12x + 5 \)

1. **Factor out \(-3\) from the \(x^2\)- and \(x\)-terms**:
   \[
   y = -3(x^2 + 4x) + 5
   \]

2. **Complete the square**:
   Take half of the coefficient of \(x\), which is \(4\), divide by 2 to get \(2\), and square it:
   \[
   \left(\frac{4}{2}\right)^2 = 4
   \]
   Add and subtract \(4\) inside the parentheses:
   \[
   y = -3(x^2 + 4x + 4 - 4) + 5
   \]
   \[
   y = -3((x + 2)^2 - 4) + 5
   \]

3. **Distribute the \(-3\)**:
   \[
   y = -3(x + 2)^2 + 12 + 5
   \]

4. **Simplify**:
   \[
   y = -3(x + 2)^2 + 17
   \]

Thus, the equation in completed square form is:
\[
y = -3(x + 2)^2 + 17
\]

---

Would you like further clarification or additional details on any of the steps? Here are five related questions you might find helpful:

1. How does completing the square help in finding the vertex of a parabola?
2. Can you use completing the square to solve quadratic equations?
3. What are the differences between completing the square and factoring quadratics?
4. How do you convert the vertex form back to standard form?
5. How does the sign of the coefficient of \(x^2\) affect the shape of the graph?

**Tip**: Completing the square is a great method for converting a quadratic equation into vertex form, which makes it easier to analyze its graph (like finding the vertex and axis of symmetry).

Complete the square for the following equations: a) y = x^2 + 10x + 12, b) y = -3x^2 - 12x + 5.

This page explains how to complete the square for the quadratic equations y = x^2 + 10x + 12 and y = -3x^2 - 12x + 5, covering essential algebraic techniques for transforming standard form into vertex form.

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