Question

factor completely.
$192n^{2}-75$

Explanation:

Step1: Factor out GCD

First, find the greatest common divisor (GCD) of 192 and 75, which is 3.
$192n^2 - 75 = 3(64n^2 - 25)$

Step2: Recognize difference of squares

Notice that $64n^2 - 25$ is a difference of squares, where $64n^2=(8n)^2$ and $25=5^2$. Use the identity $a^2 - b^2=(a-b)(a+b)$.
$64n^2 - 25 = (8n - 5)(8n + 5)$

Step3: Combine factors

Substitute the factored form back into the expression from Step1.
$3(64n^2 - 25) = 3(8n - 5)(8n + 5)$

Answer:

$3(8n - 5)(8n + 5)$