If `4^x-2^(x+2)+5+||b-1|-3|-|siny|, x , y , b in R ,` then the possible value of `b` is_________

Explanation

To solve the equation \(4^x - 2^{(x+2)} + 5 + ||b-1|-3| - |\sin y| = 0\) for the possible values of \(b\), we can follow these steps:

### Step 1: Rewrite the equation
We start by rewriting \(4^x\) in terms of \(2^x\):
\[ 4^x = (2^2)^x = (2^x)^2 \]
Thus, we can express the equation as:
\[ (2^x)^2 - 2^{(x+2)} + 5 + ||b-1|-3| - |\sin y| = 0 \]

### Step 2: Simplify the expression
Next, we simplify \(2^{(x+2)}\):
\[ 2^{(x+2)} = 2^x \cdot 2^2 = 4 \cdot 2^x \]
Now, substituting this back into the equation gives:
\[ (2^x)^2 - 4 \cdot 2^x + 5 + ||b-1|-3| - |\sin y| = 0 \]

### Step 3: Form a quadratic expression
Let \(u = 2^x\). The equation now becomes:
\[ u^2 - 4u + 5 + ||b-1|-3| - |\sin y| = 0 \]
This can be rearranged to:
\[ u^2 - 4u + (5 + ||b-1|-3| - |\sin y|) = 0 \]

### Step 4: Analyze the quadratic
For the quadratic \(u^2 - 4u + k = 0\) (where \(k = 5 + ||b-1|-3| - |\sin y|\)) to have real solutions, the discriminant must be non-negative:
\[ D = (-4)^2 - 4 \cdot 1 \cdot k \geq 0 \]
This simplifies to:
\[ 16 - 4k \geq 0 \implies k \leq 4 \]
Substituting back for \(k\):
\[ 5 + ||b-1|-3| - |\sin y| \leq 4 \]
This leads to:
\[ ||b-1|-3| - |\sin y| \leq -1 \]

### Step 5: Analyze the inequality
Since \(|\sin y|\) is always non-negative, we can conclude that:
\[ ||b-1|-3| \leq -1 + |\sin y| \]
This implies that \(||b-1|-3|\) must be at least 0, which leads to a contradiction unless \(|\sin y| = 0\).

### Step 6: Set \(|\sin y| = 0\)
Thus, we have:
\[ ||b-1|-3| \leq -1 \]
This means:
\[ ||b-1|-3| = 0 \]
From this, we can deduce:
\[ |b-1| = 3 \]

### Step 7: Solve for \(b\)
This absolute value equation gives us two cases:
1. \(b - 1 = 3 \implies b = 4\)
2. \(b - 1 = -3 \implies b = -2\)

### Conclusion
The possible values of \(b\) are:
\[ \boxed{-2 \text{ and } 4} \]