To factor the expression \(4x^2 - 9y^2 + 20x + 25\), we will follow a systematic approach. ### Step 1: Rearranging the Expression First, let's rearrange the expression to group the quadratic terms and the constant terms: \[ 4x^2 + 20x - 9y^2 + 25 \] ### Step 2: Completing the Square Next, we will complete the square for the \(x\) terms. The expression \(4x^2 + 20x\) can be factored as follows: 1. Factor out the coefficient of \(x^2\) (which is 4): \[ 4(x^2 + 5x) \] 2. To complete the square for \(x^2 + 5x\), we take half of the coefficient of \(x\) (which is 5), square it, and add and subtract it inside the parentheses: \[ \text{Half of 5 is } \frac{5}{2}, \text{ and squaring it gives } \left(\frac{5}{2}\right)^2 = \frac{25}{4}. \] Thus, we rewrite: \[ 4\left(x^2 + 5x + \frac{25}{4} - \frac{25}{4}\right) = 4\left((x + \frac{5}{2})^2 - \frac{25}{4}\right) \] 3. Distributing the 4 gives: \[ 4\left(x + \frac{5}{2}\right)^2 - 25 \] ### Step 3: Substitute Back into the Expression Now, substituting back into the expression, we have: \[ 4\left(x + \frac{5}{2}\right)^2 - 25 - 9y^2 + 25 \] The \(+25\) and \(-25\) cancel out, leaving us with: \[ 4\left(x + \frac{5}{2}\right)^2 - 9y^2 \] ### Step 4: Recognizing the Difference of Squares The expression \(4\left(x + \frac{5}{2}\right)^2 - 9y^2\) is a difference of squares, which can be factored using the formula \(a^2 - b^2 = (a - b)(a + b)\). Here, let: - \(a = 2\left(x + \frac{5}{2}\right)\) - \(b = 3y\) Thus, we can write: \[ (2\left(x + \frac{5}{2}\right) - 3y)(2\left(x + \frac{5}{2}\right) + 3y) \] ### Step 5: Simplifying the Factors Now, simplifying the factors: 1. The first factor becomes: \[ 2x + 5 - 3y \] 2. The second factor becomes: \[ 2x + 5 + 3y \] So, we can write the factored form as: \[ (2x - 3y + 5)(2x + 3y + 5) \] ### Conclusion: Correct Option The correct option is **C**: \((2x - 3y + 5)(2x + 3y + 5)\). ### Explanation of Other Options - **Option A**: \((2x - 3y + 5)(2x - 3y - 5)\) is incorrect because it does not represent the correct factors derived from the expression. - **Option B**: \((2x - 3y)(2x + 3y)\) is incorrect as it does not account for the constant term \(+5\) in the original expression. - **Option D**: \((2x + 5)(2x - 9y + 5)\) is incorrect because it does not match the structure of the factored form we derived. ### Revision Summary - Factor the quadratic expression by completing the square. - Recognize the difference of squares to factor further. - Ensure to simplify and check each factor against the original expression. - Understand why other options do not satisfy the original equation.