Coefficient

Using binomial theorem, $\newcommand{\dsum}{\displaystyle\sum} (2y - 7)^5 = \dsum_{k = 0}^5 \binom{5}k (2y)^k (-7)^{5 -k}$

To find the term in y^4, the power needs to be 4. So let k = 4:

$\text{term in }y^4 = \displaystyle\binom{5}4 (2y)^4 (-7)^1 = -560y^4$

Coefficient of y^4 = -560.