Let's solve the given equation step-by-step using substitution.

Problem

The equation given is: $(x + 2)^2 - 4(x + 2) + 3 = 0$

We are asked to use substitution, defining $z = x + 2$.

Steps to Solve

1. Substitute $z = x + 2$ into the equation:

   • This substitution turns the equation into: $z^2 - 4z + 3 = 0$

2. Solve the equation in terms of $z$:

   • Now we have a quadratic equation in $z$: $z^2 - 4z + 3 = 0$
   • To solve this quadratic equation, we can factor it: $(z - 3)(z - 1) = 0$
   • Setting each factor equal to zero gives: $z - 3 = 0 \quad \Rightarrow \quad z = 3$ $z - 1 = 0 \quad \Rightarrow \quad z = 1$

3. Back-substitute to find $x$:

   • Since $z = x + 2$, we can substitute back for each value of $z$:
     • If $z = 3$: $x + 2 = 3 \quad \Rightarrow \quad x = 3 - 2 = 1$
     • If $z = 1$: $x + 2 = 1 \quad \Rightarrow \quad x = 1 - 2 = -1$

Solution

The values of $x$ are: $x = 1 \quad \text{and} \quad x = -1$

Would you like more details on the factoring or any other part of the solution?

Here are five related questions to deepen your understanding:

1. How can you determine the correct substitution variable in similar problems?
2. What other methods can be used to solve a quadratic equation aside from factoring?
3. Why does back-substitution work when solving equations with substitution?
4. How can you verify the solutions you found for $x$ are correct?
5. Can you use this substitution technique in equations that aren’t quadratic?

Tip: Always double-check your solutions by substituting them back into the original equation to ensure they satisfy it.