You started correctly: well done.

$2^{(2 - 5x)} = 4^x \implies 2^{(2 - 5x)} = \left(2^2\right)^x \implies 2^{(2 - 5x)} = 2^{2x} \implies 2 - 5x = 2x.$ That was the hard part. Then

$2 - 5x = 2x \implies 2 - 5x + 5x = 2x + 5x \implies 2 = 2x + 5x \implies 2 = 7x \implies \dfrac{1}{7} * 7x = \dfrac{1}{7} * 2 \implies x = \dfrac{2}{7}.$

Check

$2^{(2 - 5 * \frac{2}{7})} = 2^{\frac{14}{7} - \frac{10}{7}} = 2^{\frac{4}{7}} = \sqrt[7]{2^4} = \sqrt[7]{16} = \sqrt[7]{4^2} = 4^{\frac{2}{7}}.$