`x^2-2x+(7)/(16)`

Explanation

To factorize the polynomial \( x^2 - 2x + \frac{7}{16} \), we will follow these steps:

### Step 1: Eliminate the Fraction
First, we can eliminate the fraction by multiplying the entire expression by 16 (the denominator).

\[ 16(x^2 - 2x + \frac{7}{16}) = 16x^2 - 32x + 7 \]

### Step 2: Rewrite the Quadratic
Now we have the quadratic expression \( 16x^2 - 32x + 7 \).

### Step 3: Factor by Splitting the Middle Term
Next, we need to factor this quadratic. To do this, we look for two numbers that multiply to \( 16 \times 7 = 112 \) and add to \( -32 \). The numbers that satisfy this condition are \( -28 \) and \( -4 \).

So we can rewrite the expression as:

\[ 16x^2 - 28x - 4x + 7 \]

### Step 4: Group the Terms
Now, we group the terms:

\[ (16x^2 - 28x) + (-4x + 7) \]

### Step 5: Factor Out Common Terms
Now we factor out the common terms from each group:

1. From the first group \( 16x^2 - 28x \), we can factor out \( 4x \):
\[ 4x(4x - 7) \]

2. From the second group \( -4x + 7 \), we can factor out \( -1 \):
\[ -1(4x - 7) \]

So we have:

\[ 4x(4x - 7) - 1(4x - 7) \]

### Step 6: Factor Out the Common Binomial
Now we can factor out the common binomial \( (4x - 7) \):

\[ (4x - 7)(4x - 1) \]

### Step 7: Write the Final Expression
Finally, we can write the expression as:

\[ \frac{(4x - 7)(4x - 1)}{16} \]

### Final Result
Thus, the factorization of \( x^2 - 2x + \frac{7}{16} \) is:

\[ \frac{(4x - 7)(4x - 1)}{16} \]

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