1. **State the problem**: We need to factorize and solve the quadratic equation $$x^2 + 4x - 21 = 0.$$\n\n2. **Factorize the quadratic**: Find two numbers that multiply to $$-21$$ (the constant term) and add to $$4$$ (the coefficient of $$x$$). Those numbers are $$7$$ and $$-3$$ because $$7 \times (-3) = -21$$ and $$7 + (-3) = 4$$.\n\n3. Write the quadratic as a product of two binomials:\n$$x^2 + 4x - 21 = (x + 7)(x - 3).$$\n\n4. **Solve the equation** by setting each factor equal to zero and solving for $$x$$:\n$$x + 7 = 0 \implies x = -7,$$\n$$x - 3 = 0 \implies x = 3.$$\n\n5. **Final answer**: The solutions to the quadratic equation are $$x = -7$$ and $$x = 3.$$