a. \( x^{2}+5 x+4=0 \)

To solve the quadratic equation: \[ x^{2} + 5x + 4 = 0 \] **Step 1: Identify the coefficients** The equation is in the standard quadratic form \( ax^2 + bx + c = 0 \), where: - \( a = 1 \) - \( b = 5 \) - \( c = 4 \) **Step 2: Factor the quadratic** We look for two numbers that multiply to \( a \times c = 1 \times 4 = 4 \) and add up to \( b = 5 \). The numbers **1** and **4** satisfy these conditions because: - \( 1 \times 4 = 4 \) - \( 1 + 4 = 5 \) **Step 3: Rewrite and factor by grouping** \[ x^{2} + 5x + 4 = x^2 + x + 4x + 4 \] \[ = (x^2 + x) + (4x + 4) \] \[ = x(x + 1) + 4(x + 1) \] \[ = (x + 4)(x + 1) = 0 \] **Step 4: Solve for \( x \)** Set each factor equal to zero: 1. \( x + 4 = 0 \) \( \Rightarrow x = -4 \) 2. \( x + 1 = 0 \) \( \Rightarrow x = -1 \) **Solution:** The solutions to the equation \( x^{2} + 5x + 4 = 0 \) are: \[ x = -4 \quad \text{and} \quad x = -1 \]