To find the completely factored form of the quadratic expression 16x² + 8x + 32, we will first check if there are any common factors that can be factored out from all the terms.
All the coefficients (16, 8, and 32) have a common factor of 8. We can factor out 8 from the expression:
16x² + 8x + 32 = 8(2x² + x + 4)Now, we’ll focus on factoring the quadratic expression 2x² + x + 4 further, if possible. We will use the factorization method, which involves looking for two numbers that multiply to (2 * 4) = 8 (the product of the coefficient of x² and the constant term) and add up to 1 (the coefficient of x).
However, in this case, we can see that there are no two numbers that satisfy these conditions. Additionally, we can check the discriminant of the quadratic formula b² – 4ac here:
Discriminant = (1)² - 4(2)(4) = 1 - 32 = -31The discriminant is negative, indicating that the quadratic does not factor nicely over the real numbers, and it has no real roots.
Therefore, the completely factored form of the expression remains as:
8(2x² + x + 4)In summary, the expression 16x² + 8x + 32 can be factored to 8(2x² + x + 4), but 2x² + x + 4 does not factor further in the realm of real numbers.