To solve the equation \( x^2 + 4x - 1 = 2x + 2 \), we can first rearrange it into standard form. Start by moving all terms to one side of the equation:
\[ x^2 + 4x - 1 - 2x - 2 = 0 \]
This simplifies to:
\[ x^2 + 2x - 3 = 0 \]
Now, we can factor the quadratic equation. We need to find two numbers that multiply to \(-3\) (the constant term) and add to \(2\) (the coefficient of \(x\)). The numbers that fit this are \(3\) and \(-1\). Therefore, we can factor the equation as follows:
\[ (x + 3)(x - 1) = 0 \]
Now we can find the solutions by setting each factor to zero:
- \(x + 3 = 0 \rightarrow x = -3\)
- \(x - 1 = 0 \rightarrow x = 1\)
Thus, the solutions to the equation are \(x = -3\) and \(x = 1\).
The solution set is:
\[ { -3, 1 } \]