f(x)=3x^2+2x. Find f(x-5). Write your answer without parentheses, and simplify it as much as possible. 

We are asked to find \( f(x-5) \) for the function \( f(x) = 3x^2 + 2x \). To do this, we substitute \( x-5 \) in place of \( x \) in the function:

\[
f(x-5) = 3(x-5)^2 + 2(x-5)
\]

### Step 1: Expand \( (x-5)^2 \)
\[
(x-5)^2 = x^2 - 10x + 25
\]

### Step 2: Substitute the expanded expression into the function
\[
f(x-5) = 3(x^2 - 10x + 25) + 2(x-5)
\]

### Step 3: Distribute the constants
\[
f(x-5) = 3x^2 - 30x + 75 + 2x - 10
\]

### Step 4: Simplify by combining like terms
\[
f(x-5) = 3x^2 - 28x + 65
\]

So, the simplified expression for \( f(x-5) \) is:
\[
f(x-5) = 3x^2 - 28x + 65
\]

Would you like further explanation, or have any questions about this solution?

Here are five related questions for further practice:
1. Find \( f(x+3) \) for the same function.
2. What is \( f(2x) \) for \( f(x) = 3x^2 + 2x \)?
3. If \( f(x) = 3x^2 + 2x \), what is \( f(1) \)?
4. Differentiate \( f(x) = 3x^2 + 2x \).
5. Find the vertex of the quadratic function \( f(x) = 3x^2 + 2x \).

**Tip:** Always simplify step-by-step when substituting values into functions.

f(x)=3x^2+2x. Find f(x-5). Write your answer without parentheses, and simplify it as much as possible.

Learn how to solve f(x-5) for the quadratic function f(x) = 3x^2 + 2x. Follow a step-by-step process using substitution, expansion, and simplification techniques. This problem explores key algebraic concepts, suitable for students in Grades 8-10.

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