Solve (x + 3)² + (x + 3) - 2 = 0. Let u = Rewrite the equation in terms of u. (u² + 3) + u - 2 = 0, u² + u - 2 = 0, (u² + 9) + u - 2 = 0, u² + u + 1 = 0. Factor the equation. What are the solutions of the original equation?

Solution:

Given, (x + 3)² + (x + 3) - 2 = 0

We have to find the solutions of the original equation.

Let u = x + 3

So, u² + u - 2 = 0

Factoring the equation,

u² + 2u - u - 2 = 0

u(u + 2) - 1(u + 2) = 0

(u - 1)(u + 2) = 0

u - 1 = 0

u = 1

u + 2 = 0

u = -2

Therefore, the factors are -2 and 1.

For u = -2

-2 = x + 3

x = -3 - 2

x = -5

For u = 1

1 = x + 3

x = 1 - 3

x = -2

Therefore, the solutions of the original equation are x = -2 and x = -5.

---

Solve (x + 3)2 + (x + 3) - 2 = 0. Let u = Rewrite the equation in terms of u. (u² + 3) + u - 2 = 0, u² + u - 2 = 0, (u² + 9) + u - 2 = 0, u² + u + 1 = 0. Factor the equation. What are the solutions of the original equation?

Summary:

Solving (x + 3)² + (x + 3) - 2 = 0. Let u = Rewrite the equation in terms of u. (u² + 3) + u - 2 = 0, u² + u - 2 = 0, (u² + 9) + u - 2 = 0, u² + u + 1 = 0. The solutions of the original equation are x = -2 and x = -5.