Explanation:

Let’s break it down:

- We’re given two equations:

1. \( 4x^2 + y^2 = 40 \)

2. \( xy = 6 \)

- We want to find out what \( 2x + y \) equals, and check the options.

Let’s do a quick solve:

- From \( xy = 6 \), we can say \( y = \frac{6}{x} \).

- Plug \( y \) into the first equation:

\( 4x^2 + \left(\frac{6}{x}\right)^2 = 40 \)

\( 4x^2 + \frac{36}{x^2} = 40 \)

- Multiply both sides by \( x^2 \):

\( 4x^4 + 36 = 40x^2 \)

\( 4x^4 - 40x^2 + 36 = 0 \)

Let \( u = x^2 \):

\( 4u^2 - 40u + 36 = 0 \)

- Divide by 4:

\( u^2 - 10u + 9 = 0 \)

- Factor or use quadratic formula:

Roots are \( u = 9 \) and \( u = 1 \), so \( x^2 = 9 \) or \( x^2 = 1 \).

So \( x = 3 \) or \( -3 \), and \( x = 1 \) or \( -1 \).

Let’s check each:

Case 1: \( x = 3 \), \( y = 2 \):

\( 2x + y = 2*3 + 2 = 6 + 2 = 8 \)

Case 2: \( x = 1 \), \( y = 6 \):

\( 2x + y = 2*1 + 6 = 2 + 6 = 8 \)

Other combinations either go negative or repeat the results.

So the answer is 8.

- Option 1: 6—not right.

- Option 2: 8— This is the correct answer.

- Option 3: 5—nope.

- Option 4: 4—also not correct.