To factor the expression \(32b^2 - 80b + 50\), we can first look for a common factor. The greatest common factor of the coefficients (32, -80, 50) is 2.
So we can factor out 2:
\[ 32b^2 - 80b + 50 = 2(16b^2 - 40b + 25) \]
Next, we can factor the quadratic \(16b^2 - 40b + 25\). This is a perfect square trinomial and can be factored as:
\[ 16b^2 - 40b + 25 = (4b - 5)^2 \]
So, combining these, we have:
\[ 32b^2 - 80b + 50 = 2(4b - 5)^2 \]
Now, let’s analyze the provided options:
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\(2(4b - 5)(4b - 5)\): This is equivalent to \(2(4b - 5)^2\). This is correct.
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\((4b - 5)(4b - 5)\): This is simply \((4b - 5)^2\) and does not have the factor of 2. This is not correct.
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\(2(4b - 5)^2\): This matches our derived factorization exactly. This is correct.
-
\(-2(4b - 5)^2\): This has the opposite sign of what we factored; hence, it is not equivalent. This is not correct.
The correct choices are:
- \(2(4b - 5)(4b - 5)\)
- \(2(4b - 5)^2\)