Factor the trinomial completely. \[ x^{2}+19 x+60 \]

To factor the trinomial \( x^{2} + 19x + 60 \) completely, follow these steps: 1. **Identify the coefficients:** - The quadratic is in the form \( ax^{2} + bx + c \), where: - \( a = 1 \) - \( b = 19 \) - \( c = 60 \) 2. **Find two numbers that multiply to \( ac \) (which is \( 1 \times 60 = 60 \)) and add up to \( b = 19 \).** Let's list the factor pairs of 60 and see which pair sums to 19: - \( 1 \times 60 = 60 \) and \( 1 + 60 = 61 \) (Not 19) - \( 2 \times 30 = 60 \) and \( 2 + 30 = 32 \) (Not 19) - \( 3 \times 20 = 60 \) and \( 3 + 20 = 23 \) (Not 19) - \( 4 \times 15 = 60 \) and \( 4 + 15 = 19 \) ✅ - \( 5 \times 12 = 60 \) and \( 5 + 12 = 17 \) (Not 19) - \( 6 \times 10 = 60 \) and \( 6 + 10 = 16 \) (Not 19) The pair \( 4 \) and \( 15 \) works because: \[ 4 \times 15 = 60 \quad \text{and} \quad 4 + 15 = 19 \] 3. **Write the factored form using these numbers:** \[ x^{2} + 19x + 60 = (x + 4)(x + 15) \] 4. **Verify the factoring by expanding:** \[ (x + 4)(x + 15) = x \cdot x + x \cdot 15 + 4 \cdot x + 4 \cdot 15 = x^{2} + 15x + 4x + 60 = x^{2} + 19x + 60 \] The expansion matches the original trinomial, confirming that the factorization is correct. **Final Answer:** \[ x^{2} + 19x + 60 = (x + 4)(x + 15) \]