Okay, let's see. I need to find the maximum value of the function 4(x + 7)(2 - x) for all real numbers x. Hmm, alright. Let me start by understanding the problem. So, the function is a quadratic, right? Because when you expand it, the x terms will multiply each other, resulting in an x squared term. Since the coefficient of x squared will be negative (because expanding (x + 7)(2 - x) gives -x² + ...), this quadratic opens downward, meaning it has a maximum point. So, the vertex of this parabola will give the maximum value. That makes sense.

First, maybe I should expand the expression to write it in standard quadratic form. Let me do that. Let's multiply (x + 7)(2 - x). Using the distributive property: x*2 + x*(-x) + 7*2 + 7*(-x). That would be 2x - x² + 14 - 7x. Combining like terms: (2x - 7x) + (-x²) + 14. So that's -5x - x² + 14. Then, multiply the entire expression by 4: 4*(-x² -5x +14) = -4x² -20x +56. So the quadratic is f(x) = -4x² -20x +56.

Now, since this is a quadratic function, the vertex will give the maximum value because the coefficient of x² is negative. The x-coordinate of the vertex can be found using the formula -b/(2a), where a is the coefficient of x² and b is the coefficient of x. In this case, a = -4 and b = -20. Wait, let me check that again. The standard form is ax² + bx + c. So here, a is -4, b is -20. So plugging into the formula: x = -b/(2a) = -(-20)/(2*(-4)) = 20/(-8) = -2.5. So the x-coordinate of the vertex is -2.5. Then, to find the maximum value, I need to plug this x back into the function f(x) to find f(-2.5).

Alternatively, maybe there's a quicker way without expanding. Let me think. The original function is 4(x + 7)(2 - x). Maybe I can complete the square or use another method. But since expanding gives a standard quadratic, and we already have the coefficients, maybe the vertex formula is the easiest.

So, x = -2.5. Let's compute f(-2.5). But maybe it's easier to use the original factored form. Let's see. The original function is 4(x + 7)(2 - x). Plugging x = -2.5 into this. So, (-2.5 +7) = 4.5, and (2 - (-2.5)) = 4.5. So, 4*(4.5)*(4.5) = 4*(20.25) = 81. So, the maximum value is 81? Wait, that seems straightforward. Let me verify with the expanded form as well.

Using the expanded form f(x) = -4x² -20x +56. Plugging x = -2.5. First, compute x²: (-2.5)^2 = 6.25. Then, -4*6.25 = -25. Then, -20*(-2.5) = 50. So, -25 +50 +56 = 81. Yep, same result. So that checks out.

Alternatively, maybe I can use calculus. Taking the derivative of f(x) and setting it to zero. Let's try that. The derivative of f(x) = -4x² -20x +56 is f'(x) = -8x -20. Setting this equal to zero: -8x -20 = 0. Solving for x: -8x = 20 => x = -20/8 = -2.5. Then, plugging back into f(x) gives the same maximum value of 81. So, three different methods all lead to the same answer. That seems solid.

Wait, but just to make sure I didn't make any arithmetic errors. Let me go through the steps again. Starting with the original function: 4(x +7)(2 -x). Let's expand (x +7)(2 -x). First term: x*2 = 2x. Then x*(-x) = -x². Then 7*2 = 14. Then 7*(-x) = -7x. So combining terms: 2x -x² +14 -7x. Combine like terms: (2x -7x) = -5x. So -x² -5x +14. Multiply by 4: -4x² -20x +56. Correct. Then, vertex at x = -b/(2a) = -(-20)/(2*(-4)) = 20/-8 = -2.5. Then, plugging into the original function: (x +7)(2 -x) when x = -2.5: (-2.5 +7) = 4.5, (2 - (-2.5)) = 4.5. 4.5*4.5 = 20.25, multiplied by 4 gives 81. All steps check out.

Alternatively, using the vertex formula for the quadratic. For any quadratic ax² + bx + c, the maximum (if a < 0) is at x = -b/(2a). Then, the maximum value is c - b²/(4a). Wait, is that right? Let me recall. The formula for the vertex value is f(-b/(2a)) = c - b²/(4a). Let me verify. Let's take the expanded quadratic: -4x² -20x +56. So a = -4, b = -20, c =56. Then, the y-coordinate of the vertex is c - b²/(4a). Plugging in: 56 - (-20)^2/(4*(-4)) = 56 - 400/(-16) = 56 + 25 = 81. Yep, same result. So that's another way to compute it without plugging back into the function. That's a good formula to remember.

So, all methods confirm that the maximum value is 81. Therefore, the answer should be 81. Let me just check once more with another approach. Suppose I complete the square on the expanded quadratic. Let's see.

Starting with f(x) = -4x² -20x +56. Factor out -4 from the first two terms: -4(x² +5x) +56. Now, to complete the square inside the parentheses: x² +5x. Take half of 5, which is 2.5, square it: 6.25. So, x² +5x +6.25 -6.25 = (x +2.5)^2 -6.25. Therefore, the expression becomes -4[(x +2.5)^2 -6.25] +56 = -4(x +2.5)^2 +25 +56 = -4(x +2.5)^2 +81. Since the square term is always non-negative, the maximum value occurs when the square term is zero, which is at x = -2.5, giving the maximum value of 81. Yep, that works too.

So, all methods converge to 81. I think that's solid. I don't see any mistakes in the calculations. The key was expanding the original quadratic, recognizing it's a downward-opening parabola, and then using the vertex formula. Alternatively, using calculus or completing the square gives the same result. So, I feel confident that 81 is the correct maximum value.
Therefore, the maximum value of $4(x + 7)(2 - x)$ is $\boxed{81}$.
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