The maximum value of 4 sin2x + 3 cos2x is :

y = 4 sin²x + 3 cos²x

y = sin²x + 3(sin²x + cos²x)

y = sin²x + 3

since sin²x + cos²x = 1

y = sin²x + 3

we know that  0 ≤ sin²x ≤ 1

so   3 ≤ sin²x + 3 ≤ 4

so y_max = 4