To find the common factor of the expressions \(25x^2 + 40x + 16\) and \(25x^2 - 16\), let's analyze each expression.
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Factor \(25x^2 + 40x + 16\): This expression is a quadratic trinomial. We can factor it by finding two numbers that multiply to \(25 \times 16 = 400\) and add to \(40\). The numbers \(20\) and \(20\) work, so we can factor it as follows:
\[ 25x^2 + 40x + 16 = (5x + 4)(5x + 4) = (5x + 4)^2 \]
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Factor \(25x^2 - 16\): This expression is a difference of squares. It can be factored using the formula \(a^2 - b^2 = (a - b)(a + b)\):
\[ 25x^2 - 16 = (5x)^2 - 4^2 = (5x - 4)(5x + 4) \]
Now, let's identify the common factor between the two factored forms:
- From \(25x^2 + 40x + 16\), we have \((5x + 4)^2\).
- From \(25x^2 - 16\), we have \((5x - 4)(5x + 4)\).
The common factor in these two expressions is \((5x + 4)\).