To find the completely factored form of the expression 16x² + 8x + 32, we first start by looking for common factors and then simplifying the expression.
1. **Identify Common Factors**: The coefficients of the terms are 16, 8, and 32. The greatest common factor (GCF) of these numbers is 8.
2. **Factor out the GCF**: We can factor 8 out of each term in the expression:
16x² + 8x + 32 = 8(2x² + x + 4)
3. **Factor the Quadratic Expression**: Now, we need to factor the quadratic inside the parentheses, 2x² + x + 4. To do this, we look for two numbers that multiply to (2 * 4 = 8) and add up to 1 (the coefficient of x). However, there are no pairs of real numbers that satisfy these requirements. Thus, the quadratic does not factor nicely into real numbers.
Thus, we conclude that:
16x² + 8x + 32 = 8(2x² + x + 4)
In conclusion, the completely factored form of the expression is 8(2x² + x + 4), where 2x² + x + 4 does not reduce further with real coefficients.