Determine if the point-estimation is unbiased [Bias of an estimator]

Question

The problem:

Let x₁, x₂, x₃, ..., x₇ be observations of the independent random variables X₁, X₂, X₃, ..., X₇, such that E(X_i) = μ and Var(X_i) = σ²

Let
σobs2 = c(x22 + x62 - ½((x1 + x7)2)
Be the point-estimation of σ²

Determine the constant c so that σobs2 becomes unbiased.

Here is my attempt:
E(σ_obs²) = cE(x₂² + x₆² - ½((x₁ + x₇)²)) = c(E(x₂²) + E(x₆²) - E(½((x₁ + x₇)²)) =
c(E(x₂²) + E(x₆²) - ½E((x₁ + x₇)²))) = c(E(x₂²) + E(x₆²) - ½E(x₁² - 2x₁x₇² + x₇²)) =
c(E(x₂²) + E(x₆²) - ½(E(x₁²) -2E(x₁x₇) + E(x₇²))) = ...

This is where I get stuck, is there anyone who can give me a hint? I am trying to get E(σ_obs²) = σ²
I am really thankful for any help or advice on this.

Update:
E(σ_obs²) = cE(x₂² + x₆² - ½((x₁ + x₇)²)) = c(E(x₂²) + E(x₆²) - E(½((x₁ + x₇)²)) =
c(E(x₂²) + E(x₆²) - ½E((x₁ + x₇)²))) = c(E(x₂²) + E(x₆²) - ½E(x₁² - 2x₁x₇² + x₇²)) =
c(E(x₂²) + E(x₆²) - ½(E(x₁²) -2E(x₁x₇) + E(x₇²))) =
c(E(x₂²) + E(x₆²) - ½(E(x₁²) -2E(x₁)E(x₇) + E(x₇²))) =
c(E(x₂²) + E(x₆²) - ½E(x₁²) + E(x₁)E(x₇) - ½E(x₇²)) =
c(E(x₂²) + E(x₆²) - ½E(x₁²) + E(x₁)E(x₇) - ½E(x₇²)) =
c(V(x₂) + E²(x₂) + V(x₆) + E²(x₆) - ½V(x₁) + -½E²(x₁) + E(x₁)E(x₇) - ½V(x₇) - ½ E²(x₇)) =
c(V(x₂) + V(x₆) - ½V(x₁) - ½V(x₇) + E²(x₂) + E²(x₆) -½E²(x₁) - ½ E²(x₇) + E(x₁)E(x₇)) =
c(σ² + σ² - ½σ² - ½σ² + μ² + μ² - ½μ² - ½ μ² + μ²)=
c(σ² + σ² - σ² + μ² + μ² - μ² + μ²) =
c(σ² + μ² + μ²) =
c(σ² + 2μ²)