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You are asked to express the square of the trinomial \( 4x^2 + 20x + 30 \) in the form \( (ax + b)^2 + c \) using the formula \( a^2x^2 + 2abx + b^2 + c \).

### Solution:

Let's break it down step by step:

We are working with:
\[ 4x^2 + 20x + 30 \]

1. **Factor the quadratic part**: 
   The quadratic term \( 4x^2 \) suggests that \( a^2 = 4 \), so \( a = 2 \).
   
2. **Rewrite the middle term**: 
   The middle term is \( 20x \), and it corresponds to \( 2abx \) in the formula. Given that \( a = 2 \), we have:
   \[ 2ab = 20 \]
   Solving for \( b \), we get:
   \[ 2 \cdot 2 \cdot b = 20 \quad \Rightarrow \quad b = 5 \]

3. **Find \( b^2 \)**:
   Now, calculate \( b^2 \):
   \[ b^2 = 5^2 = 25 \]

4. **Compare with the constant term**:
   The original constant term is 30, and in the expansion, we would have \( b^2 + c \). Since \( b^2 = 25 \), we find \( c \) by solving:
   \[ b^2 + c = 30 \quad \Rightarrow \quad 25 + c = 30 \quad \Rightarrow \quad c = 5 \]

### Final answer:
Thus, the expression can be written as:
\[
(2x + 5)^2 + 5
\]

Would you like further clarification or more details on this?

Here are 5 related questions to explore further:
1. How would you handle a trinomial with negative coefficients?
2. What is the geometric meaning of completing the square?
3. How can this method be extended to solve quadratic equations?
4. Can you factor the trinomial without using the formula?
5. How does completing the square relate to the quadratic formula?

**Tip**: Whenever you square a binomial, always remember the general form \( (ax + b)^2 = a^2x^2 + 2abx + b^2 \) to check your work!

Find the square of 4x^2 + 20x + 30 in the form (ax + b)^2 + c using the formula a^2x^2 + 2abx + b^2 + c.

Learn how to express the trinomial 4x^2 + 20x + 30 in the form (ax + b)^2 + c by completing the square. Step-by-step breakdown of solving this quadratic expression. Suitable for Grades 9-10.

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