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The equation given is:

\[
x^2 + 7x - 2 = (x + a)^2 + b
\]

We need to find the values of \( a \) and \( b \).

### Step 1: Expand \( (x + a)^2 \)

Expanding \( (x + a)^2 \):

\[
(x + a)^2 = x^2 + 2ax + a^2
\]

So the equation becomes:

\[
x^2 + 7x - 2 = x^2 + 2ax + a^2 + b
\]

### Step 2: Compare the terms

We can now compare the coefficients of both sides of the equation. On the left-hand side, we have:

\[
x^2 + 7x - 2
\]

On the right-hand side, after expanding:

\[
x^2 + 2ax + (a^2 + b)
\]

Now, compare the coefficients of the corresponding powers of \( x \):

1. **Coefficient of \( x^2 \)**: On both sides, the coefficient of \( x^2 \) is 1, so this doesn't give us any new information.
   
2. **Coefficient of \( x \)**: The coefficient of \( x \) on the left side is 7, and on the right side, it is \( 2a \). Therefore, we have the equation:

   \[
   2a = 7 \quad \Rightarrow \quad a = \frac{7}{2}
   \]

3. **Constant term**: The constant term on the left side is \(-2\), and on the right side, it is \( a^2 + b \). So, we have:

   \[
   a^2 + b = -2
   \]

### Step 3: Solve for \( b \)

Substitute \( a = \frac{7}{2} \) into \( a^2 + b = -2 \):

\[
\left( \frac{7}{2} \right)^2 + b = -2
\]

\[
\frac{49}{4} + b = -2
\]

\[
b = -2 - \frac{49}{4}
\]

\[
b = \frac{-8}{4} - \frac{49}{4} = \frac{-57}{4}
\]

### Final Answer:

The values of \( a \) and \( b \) are:

\[
a = \frac{7}{2}, \quad b = \frac{-57}{4}
\]

---

Would you like further explanation on any step or details about solving similar problems?

Here are five related questions to explore:

1. How do you expand expressions like \( (x + a)^2 \)?
2. What is the method of comparing coefficients in equations?
3. How do you solve for unknowns when dealing with quadratic equations?
4. Can you explain the process of expanding binomials?
5. How do constant terms influence the solution of an equation?

**Tip:** When solving equations by comparing coefficients, always align similar powers of \( x \) to set up simpler equations for unknowns.

x^2 + 7x - 2 = (x + a)^2 + b. Find the value of a and b.

Learn how to solve the quadratic equation x^2 + 7x - 2 = (x + a)^2 + b by expanding and comparing coefficients. This problem demonstrates key algebraic concepts, including expanding binomials and solving for unknowns. Suitable for students in Grades 9-11.

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