What can be added to \( f(x)=2 x^{2}-3 x f(x)=2 x^{2}-3 x \) so that \( (2 x-7)(2 x-7) \) and \( (x+5)(x+5) \) are factors of \( f(x) f(x) \) ?

To determine what needs to be added to the given function \( f(x) = 2x^2 - 3x \) so that \( (2x-7)^2 \) and \( (x+5)^2 \) are factors of \( f(x)^2 \), follow these steps: ### Step 1: Understand the Implications If \( (2x-7)^2 \) and \( (x+5)^2 \) are factors of \( f(x)^2 \), then \( f(x) \) itself must have \( (2x-7) \) and \( (x+5) \) as factors. This is because squaring \( f(x) \) would square each of its factors. ### Step 2: Construct the Desired \( f(x) \) The factors \( (2x-7) \) and \( (x+5) \) multiply to give: \[ (2x-7)(x+5) = 2x^2 + 3x - 35 \] Thus, the desired form of \( f(x) \) should be: \[ f(x) = 2x^2 + 3x - 35 \] ### Step 3: Determine What to Add The original \( f(x) \) is \( 2x^2 - 3x \). To convert it to the desired form: \[ 2x^2 - 3x + \text{(something)} = 2x^2 + 3x - 35 \] Subtracting the original \( f(x) \) from the desired form: \[ \text{Something} = (2x^2 + 3x - 35) - (2x^2 - 3x) = 6x - 35 \] ### Conclusion You need to add \( 6x - 35 \) to the original function \( f(x) = 2x^2 - 3x \) to make \( (2x-7)^2 \) and \( (x+5)^2 \) factors of \( f(x)^2 \). **Answer:** Add the linear polynomial six x minus thirty-five, that is, add 6 x − 35.