1. **Problem:** Solve by factorization the quadratic equation $$x^2 + 5x + 4 = 0$$. 2. **Formula and rules:** To solve a quadratic equation by factorization, we express it as a product of two binomials equal to zero: $$ (x + m)(x + n) = 0 $$ where $$m$$ and $$n$$ satisfy $$m + n = b$$ and $$mn = c$$ for equation $$x^2 + bx + c = 0$$. 3. **Step-by-step solution:** - The equation is $$x^2 + 5x + 4 = 0$$. - We look for two numbers whose sum is 5 and product is 4. - These numbers are 4 and 1 because $$4 + 1 = 5$$ and $$4 \times 1 = 4$$. - So, factorize as $$ (x + 4)(x + 1) = 0 $$. 4. **Find solutions:** - Set each factor equal to zero: - $$x + 4 = 0 \Rightarrow x = -4$$ - $$x + 1 = 0 \Rightarrow x = -1$$ 5. **Answer:** The solutions are $$x = -4$$ and $$x = -1$$. This method works because if the product of two factors is zero, at least one of the factors must be zero.