Find the maximum value of `4sin^2x+3cos^2x+sin(x/2)+cos(x/2)dot`

Explanation

To find the maximum value of the expression \(4\sin^2 x + 3\cos^2 x + \sin\left(\frac{x}{2}\right) + \cos\left(\frac{x}{2}\right)\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ 4\sin^2 x + 3\cos^2 x + \sin\left(\frac{x}{2}\right) + \cos\left(\frac{x}{2}\right) \]
We can use the identity \(\cos^2 x = 1 - \sin^2 x\) to rewrite \(\cos^2 x\):
...