https://file.aiquickdraw.com/mathbot/file/17267834636422by7guvr.jpg

Let's express \( x^2 + 20x + 155 \) in the form \( (x + b)^2 + c \), using the given identity as a guide.

We know from the image:

\[
x^2 + 20x + 140 = (x + 10)^2 + 40
\]

### Step 1: Compare the quadratic terms
The quadratic term is \( x^2 + 20x \), which remains the same in both expressions.

### Step 2: Completing the square
To write \( x^2 + 20x + 155 \) in the desired form, we need to complete the square.

1. Take the coefficient of \( x \), which is 20, and divide by 2: 
   \[
   \frac{20}{2} = 10
   \]

2. Square this result:
   \[
   10^2 = 100
   \]

So, \( x^2 + 20x \) can be written as:
\[
(x + 10)^2 - 100
\]

### Step 3: Adjust the constant term
Now, let's express the original expression \( x^2 + 20x + 155 \):
\[
x^2 + 20x + 155 = (x + 10)^2 - 100 + 155
\]
Simplifying the constants:
\[
(x + 10)^2 - 100 + 155 = (x + 10)^2 + 55
\]

### Final answer:
\[
x^2 + 20x + 155 = (x + 10)^2 + 55
\]

---

Do you want any further clarifications or additional details?

### Relative questions:
1. How do we determine the value of \( b \) when completing the square?
2. Why is it necessary to add and subtract \( 100 \) when completing the square?
3. Can you apply this method to a quadratic expression with a different linear term?
4. What is the general formula for completing the square in any quadratic expression?
5. How does this method relate to solving quadratic equations?

### Tip:
When completing the square, always remember to balance the equation by adding and subtracting the square of half the linear coefficient.

Express x^2 + 20x + 155 in the form (x + b)^2 + c, using the given fact x^2 + 20x + 140 = (x + 10)^2 + 40.

Learn how to express the quadratic expression x^2 + 20x + 155 in the form (x + b)^2 + c by completing the square. This solution builds on the fact that x^2 + 20x + 140 can be written as (x + 10)^2 + 40. Ideal for students in Grades 8-10 learning quadratic transformations.

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