The given function is:

$f(x+3) = 2x^2 - 3x + 1$

We are asked to find $f(x+1)$.

Step-by-step solution:

1. Substitute $x+1$ into the expression for $f(x+3)$. Notice that $x+3 = (x+1) + 2$, so we can rewrite the function by replacing $x$ with $x-2$ (because $(x+1) - 2 = x-1$).

$f(x+1) = 2(x-2)^2 - 3(x-2) + 1$

2. Now, expand the terms:

$f(x+1) = 2[(x-2)^2] - 3(x-2) + 1$ $(x-2)^2 = x^2 - 4x + 4$ $f(x+1) = 2(x^2 - 4x + 4) - 3(x - 2) + 1$

3. Simplify the expression:

$f(x+1) = 2x^2 - 8x + 8 - 3x + 6 + 1$ $f(x+1) = 2x^2 - 11x + 15$

Thus, the required function is:

$f(x+1) = 2x^2 - 11x + 15$

Would you like a more detailed breakdown, or do you have any further questions?

Here are 5 related questions you might find useful:

1. How would you solve $f(x+2)$ using the same method?
2. Can you apply the same substitution technique to a different type of function?
3. What happens if you try to find $f(x)$ from $f(x+3)$?
4. How do you expand $(x-2)^2$ systematically in algebra?
5. Can this method be applied to other function transformations, such as shifts and stretches?

Tip: When substituting into a function, always adjust the variable properly by analyzing the new terms relative to the original equation.