Simplify the expression \( (x-4)\left(x^{2}+5 x+3\right) \) by multiplying the two polynomials. Answer:

To simplify the expression \( (x - 4)(x^{2} + 5x + 3) \), follow these steps: 1. **Distribute \( x \) across the second polynomial:** \[ x \cdot x^{2} = x^{3} \] \[ x \cdot 5x = 5x^{2} \] \[ x \cdot 3 = 3x \] So, \[ x(x^{2} + 5x + 3) = x^{3} + 5x^{2} + 3x \] 2. **Distribute \( -4 \) across the second polynomial:** \[ -4 \cdot x^{2} = -4x^{2} \] \[ -4 \cdot 5x = -20x \] \[ -4 \cdot 3 = -12 \] So, \[ -4(x^{2} + 5x + 3) = -4x^{2} - 20x - 12 \] 3. **Combine the results from steps 1 and 2:** \[ x^{3} + 5x^{2} + 3x - 4x^{2} - 20x - 12 \] 4. **Combine like terms:** \[ x^{3} + (5x^{2} - 4x^{2}) + (3x - 20x) - 12 = x^{3} + x^{2} - 17x - 12 \] **Final Simplified Expression:** \[ x^{3} + x^{2} - 17x - 12 \]