Factor out the a value from the first two terms.

$f\left(x\right) = 2\left(x^{2} − 8x\right) + 34$

Complete the square. Why is $32$ subtracted?

$f\left(x\right) = 2\left(x^{2} − 8x + 16\right) + 2$

$f\left(x\right) = 2\left(x − 4\right)^{2} + 2$