Question

factor completely. 32n² - 50 (16n - 25)(16n + 25) 2(8n - 25)(8n + 25) not factorable 2(4n + 5)(4n - 5)

Explanation:

Step1: Factor out GCF

First, find the greatest common factor (GCF) of \(32n^2\) and \(50\). The GCF of 32 and 50 is 2. So, factor out 2:
\(32n^2 - 50 = 2(16n^2 - 25)\)

Step2: Apply difference of squares

Notice that \(16n^2 - 25\) is a difference of squares, since \(16n^2=(4n)^2\) and \(25 = 5^2\). The formula for difference of squares is \(a^2 - b^2=(a - b)(a + b)\). Here, \(a = 4n\) and \(b = 5\), so:
\(16n^2 - 25=(4n - 5)(4n + 5)\)

Step3: Combine factors

Substitute back into the expression from Step1:
\(2(16n^2 - 25)=2(4n - 5)(4n + 5)\) Wait, wait—wait, no, wait. Wait, the option given is \(2(4n + 5)(4n - 5)\) (which is the same as \(2(4n - 5)(4n + 5)\)). Wait, let's check the options. The last option is \(2(4n + 5)(4n - 5)\). Let's verify by expanding:

Expand \(2(4n + 5)(4n - 5)\):
First, \((4n + 5)(4n - 5)=16n^2 - 25\) (difference of squares), then multiply by 2: \(2(16n^2 - 25)=32n^2 - 50\), which matches the original expression.

Wait, let's check the options again. The options are:

1. \((16n - 25)(16n + 25)\) – expanding this gives \(256n^2 - 625\), not \(32n^2 - 50\).
2. \(2(8n - 25)(8n + 25)\) – expanding gives \(2(64n^2 - 625)=128n^2 - 1250\), not matching.
3. Not Factorable – incorrect, since we factored it.
4. \(2(4n + 5)(4n - 5)\) – expanding gives \(2(16n^2 - 25)=32n^2 - 50\), which is correct.

Answer:

\(2(4n + 5)(4n - 5)\) (the last option, \(2(4n + 5)(4n - 5)\))