Final answer:
The function f(x) = 4x² + 8x + 5, when expressed in transformation form, is 4(x + 1)² + 1 after completing the square. The provided options do not exactly match the transformed equation. Option a) is the closest, but it lacks the constant term +1.
Step-by-step explanation:
The goal is to express f(x) = 4x² + 8x + 5 in transformation form, which is a way to rewrite a quadratic equation in a form that makes it easy to understand the transformation of the function from the standard quadratic function f(x) = x². We do this by completing the square.
To complete the square:
- Rewrite the quadratic and linear terms: 4x² + 8x.
- Factor out the coefficient of x²: 4(x² + 2x).
- Add and subtract the square of half the coefficient of x inside the parentheses: 4((x + 1)² - 1).
- Add the adjustment outside the parentheses: 4((x + 1)² - 1) + 5.
- Simplify the constant terms: 4(x + 1)² + 1.
Therefore, the correct transformation form of f(x) is 4(x + 1)² + 1, which is not an exact match to any of the given options. The closest option is a) f(x) = 4(x + 1)²; however, note that our transformed function includes an additional constant term of +1, which is missing from option a).