Factorize 25a² - 4b² + 28bc - 49c².

To factor the expression \(25a^2 - 4b^2 + 28bc - 49c^2\), we can start by rearranging it for clarity:

Rearranging the Expression

The expression can be rewritten as:

\(25a^2 - (4b^2 - 28bc + 49c^2)\)

Identifying the Quadratic Form

Next, we focus on the quadratic part, \(4b^2 - 28bc + 49c^2\). This can be factored as:

\((2b - 7c)^2\)

Putting It All Together

Now, substituting back, we have:

\(25a^2 - (2b - 7c)^2\)

Applying the Difference of Squares

This expression is a difference of squares, which can be factored using the formula \(x^2 - y^2 = (x - y)(x + y)\). Here, \(x = 5a\) and \(y = (2b - 7c)\):

• First factor: \(5a - (2b - 7c) = 5a - 2b + 7c\)
• Second factor: \(5a + (2b - 7c) = 5a + 2b - 7c\)

Final Factored Form

The completely factored form of the original expression is:

\((5a - 2b + 7c)(5a + 2b - 7c)\)