Given f, of, x, equals, 3, x, squared, plus, x*f*(*x*)=3*x*2+*x*, find f, of, minus, 5*f*(−5)

Let's solve the problem step by step to find \( f(-5) \).

### Step 1: Given Equation
The equation provided is:

\[
f(x) + x \cdot f(x) = 3x^2 + x
\]

### Step 2: Factorize the Left Side
We can factor out \( f(x) \) on the left-hand side:

\[
f(x)(1 + x) = 3x^2 + x
\]

### Step 3: Solve for \( f(x) \)
To isolate \( f(x) \), divide both sides by \( (1 + x) \):

\[
f(x) = \frac{3x^2 + x}{1 + x}
\]

### Step 4: Simplify the Expression
Simplify the fraction:

\[
f(x) = \frac{x(3x + 1)}{x + 1}
\]

We can simplify this further by performing polynomial division:

\[
f(x) = 3x - 2 + \frac{2}{x + 1}
\]

So,

\[
f(x) = 3x - 2 + \frac{2}{x + 1}
\]

### Step 5: Find \( f(-5) \)
Now, substitute \( x = -5 \) into the simplified equation:

\[
f(-5) = 3(-5) - 2 + \frac{2}{-5 + 1}
\]

\[
f(-5) = -15 - 2 + \frac{2}{-4}
\]

\[
f(-5) = -17 - \frac{1}{2} = -\frac{34}{2} - \frac{1}{2} = -\frac{35}{2}
\]

So, \( f(-5) = -\frac{35}{2} \).

### Final Answer:
The value of \( f(-5) \) is \(\boxed{-\frac{35}{2}}\).

Would you like more details or have any questions?

Here are 5 related questions:
1. How would you solve \( f(x) = 0 \) for the given function \( f(x) \)?
2. What is the derivative of the function \( f(x) = 3x - 2 + \frac{2}{x+1} \)?
3. How does the function \( f(x) \) behave as \( x \) approaches \( -1 \)?
4. What is the value of \( f(2) \) using the derived function?
5. Can you find the integral of the function \( f(x) = 3x - 2 + \frac{2}{x+1} \)?

**Tip:** When simplifying fractions, consider polynomial division if the numerator and denominator have common factors.

Learn how to solve the equation f(x) + x * f(x) = 3x^2 + x step-by-step. Understand polynomial division and fractional expressions in algebra. Suitable for high school students studying advanced algebra.

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