Solve ` 2^(x+2)-2^(x+3) -2^(x+4) gt 5^(x+1) -5^(x+2)`.

Explanation

To solve the inequality \( 2^{(x+2)} - 2^{(x+3)} - 2^{(x+4)} > 5^{(x+1)} - 5^{(x+2)} \), we will simplify both sides step by step.

### Step 1: Simplify the left side
We can factor out \( 2^x \) from the left side:

\[ 2^{(x+2)} - 2^{(x+3)} - 2^{(x+4)} = 2^x \cdot 2^2 - 2^x \cdot 2^3 - 2^x \cdot 2^4 \]

This simplifies to:

\[ 2^x (4 - 8 - 16) \]

### Step 2: Simplify the left side further
Now, calculate the expression inside the parentheses:

\[ 4 - 8 - 16 = 4 - 24 = -20 \]

So, the left side becomes:

\[ 2^x \cdot (-20) \]

### Step 3: Simplify the right side
Now, let's simplify the right side:

\[ 5^{(x+1)} - 5^{(x+2)} = 5^x \cdot 5 - 5^x \cdot 5^2 \]

Factoring out \( 5^x \):

\[ 5^x (5 - 25) \]

### Step 4: Simplify the right side further
Now, calculate the expression inside the parentheses:

\[ 5 - 25 = -20 \]

So, the right side becomes:

\[ 5^x \cdot (-20) \]

### Step 5: Set up the inequality
Now we can rewrite the original inequality:

\[ 2^x \cdot (-20) > 5^x \cdot (-20) \]

### Step 6: Divide both sides by -20
Since we are dividing by a negative number, we must reverse the inequality:

\[ 2^x < 5^x \]

### Step 7: Rewrite the inequality
We can rewrite this as:

\[ \frac{2^x}{5^x} < 1 \]

This can be simplified to:

\[ \left(\frac{2}{5}\right)^x < 1 \]

### Step 8: Analyze the inequality
Since \( \frac{2}{5} < 1 \), the inequality \( \left(\frac{2}{5}\right)^x < 1 \) holds true when \( x > 0 \).

### Final Answer
Thus, the solution to the inequality is:

\[ x < 0 \]