To solve the problem, we need to follow these steps:
- Expand the expression \((2x - 7)^2\).
- Subtract this expanded polynomial from 4.
- Simplify the result to get it into standard form.
Step 1: Expand \((2x - 7)^2\)
Using the formula \((a - b)^2 = a^2 - 2ab + b^2\), we get:
\[ (2x - 7)^2 = (2x)^2 - 2 \cdot (2x) \cdot 7 + 7^2 \]
Calculating each part:
- \((2x)^2 = 4x^2\)
- \(-2 \cdot (2x) \cdot 7 = -28x\)
- \(7^2 = 49\)
Putting it all together:
\[ (2x - 7)^2 = 4x^2 - 28x + 49 \]
Step 2: Subtract this from 4
Now we subtract \(4x^2 - 28x + 49\) from 4:
\[ 4 - (4x^2 - 28x + 49) \]
This can be rewritten as:
\[ 4 - 4x^2 + 28x - 49 \]
Step 3: Simplify the expression
Now we combine like terms:
\[ 4 - 49 = -45 \]
Thus, we have:
\[ -4x^2 + 28x - 45 \]
Finally, writing it in standard form gives:
\[ -4x^2 + 28x - 45 \]
So the result when \((2x - 7)^2\) is subtracted from 4 is:
\[ \boxed{-4x^2 + 28x - 45} \]
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