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.