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 $$ f(x) f(x) $$ ?

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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.
You need to add $$ 6x - 35 $$ to $$ f(x) = 2x^2 - 3x $$ to make $$ (2x-7)^2 $$ and $$ (x+5)^2 $$ factors of $$ f(x)^2 $$.

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 \( f(x) f(x) \) ?

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