The problem asks us to find the values of $a$ and $b$ such that:
$x^2 + 2x - 7 = (x + a)^2 + b.$

Step-by-step Solution:

1. Expand the right-hand side: Expand $(x + a)^2$: $(x + a)^2 = x^2 + 2ax + a^2.$ Substituting this into the equation: $x^2 + 2x - 7 = x^2 + 2ax + a^2 + b.$

2. Simplify by matching terms: Subtract $x^2$ from both sides: $2x - 7 = 2ax + a^2 + b.$ Compare the coefficients of $x$ and the constant terms on both sides.

3. Match the coefficients:

   • For the $x$-terms: $2 = 2a$.
     Solving for $a$:
     $a = 1.$
   • For the constant terms: $-7 = a^2 + b$.
     Substitute $a = 1$:
     $-7 = 1^2 + b \implies -7 = 1 + b \implies b = -8.$

Final Answer:

$a = 1, \, b = -8.$

Would you like me to clarify any part of this solution or expand on related concepts?

Related Questions:

1. How do you expand a binomial like $(x + a)^2$?
2. Why do we match coefficients in this type of equation?
3. What other methods can be used to solve similar equations?
4. Can $a$ or $b$ take on multiple values in other problems?
5. How does this equation relate to completing the square?

Tip:

When solving for unknown coefficients, always organize terms and compare coefficients systematically to avoid errors.