Question

which function is equivalent to 2(x + 4)² + 6?
○ 2x² + 16x + 38
○ x² + 8x + 16
○ 2x² + 8x + 38
○ 2x² + 16x + 32

Explanation:

Step1: Expand \((x + 4)^2\)

Using the formula \((a + b)^2 = a^2 + 2ab + b^2\), where \(a = x\) and \(b = 4\), we get \((x + 4)^2 = x^2 + 8x + 16\).

Step2: Multiply by 2

Multiply the expanded form by 2: \(2(x^2 + 8x + 16)=2x^2 + 16x + 32\).

Step3: Add 6

Now add 6 to the result: \(2x^2 + 16x + 32+6 = 2x^2 + 16x + 38\). Wait, no, wait. Wait the original function is \(2(x + 4)^2+6\)? Wait, no, looking back at the problem: "Which function is equivalent to \(2(x + 4)^2+6\)? Wait, no, the user's image: let's re - check. Wait the original expression: the problem is "Which function is equivalent to \(2(x + 4)^2+6\)? Wait, no, maybe I misread. Wait the options: let's see the first option is \(2x^2 + 16x + 38\)? Wait no, the first option in the image: the first option is \(2x^2+16x + 38\)? Wait no, let's re - do the expansion correctly.

Wait, correct steps:

Step1: Expand \((x + 4)^2\)

\((x + 4)^2=x^{2}+8x + 16\) (by \((a + b)^2=a^{2}+2ab + b^{2}\), \(a = x\), \(b = 4\), so \(2ab = 8x\), \(b^{2}=16\))

Step2: Multiply by 2

\(2(x^{2}+8x + 16)=2x^{2}+16x + 32\)

Step3: Add 6? Wait no, wait the original expression: the problem says "equivalent to \(2(x + 4)^2+6\)"? Wait no, maybe the original expression is \(2(x + 4)^2+6\)? Wait no, looking at the options, the first option is \(2x^2 + 16x + 38\) (since \(32 + 6=38\)). Wait, let's check the options again. The first option (from the image) is \(2x^2+16x + 38\)? Wait the options are:

1. \(2x^2 + 16x + 38\)

1. \(x^2+8x + 16\)

1. \(2x^2+8x + 38\)

1. \(2x^2+16x + 32\)

Wait, let's do the expansion of \(2(x + 4)^2+6\):

First, \((x + 4)^2=x^{2}+8x + 16\)

Then, \(2(x^{2}+8x + 16)=2x^{2}+16x + 32\)

Then, \(2x^{2}+16x + 32+6=2x^{2}+16x + 38\)

So the equivalent function is \(2x^2 + 16x + 38\) (assuming the first option is \(2x^2 + 16x + 38\))

Answer:

\(2x^2 + 16x + 38\) (assuming this is the first option in the multiple - choice, e.g., if the first option is A. \(2x^2 + 16x + 38\), then the answer is A. \(2x^2 + 16x + 38\))