Explanation

Square and add: (3cos θ + 4sin θ)² + (4cos θ + 3sin θ)² = 25 + 48sin θcos θ. With 3cos θ + 4sin θ = 4, solving together with cos²θ + sin²θ = 1 gives sin θ = 7/25 (the alternative sin θ = 1 would make θ = π/2, excluded) and cos θ = 24/25. Then 4cos θ + 3sin θ = 96/25 + 21/25 = 117/25, option A.