Okay, let's see. The problem is: If f(x) = a x⁴ - b x² + x + 5 and f(-3) = 2, then what is the value of f(3)? Hmm. Alright, so we need to find f(3) given that f(-3) is 2. 

First, I should probably figure out the relationship between f(3) and f(-3). Let me write down the function again. f(x) = a x⁴ - b x² + x + 5. So, when I plug in x = -3, I get f(-3) = a*(-3)^4 - b*(-3)^2 + (-3) + 5. And that's equal to 2. Then, plugging in x = 3, f(3) would be a*(3)^4 - b*(3)^2 + 3 + 5. 

Wait, maybe there's a way to relate f(3) and f(-3) without solving for a and b. Let me compute both f(3) and f(-3) and see if there's a pattern or something that cancels out. 

Let me compute f(3) first. f(3) = a*(3)^4 - b*(3)^2 + 3 + 5. Simplifying, 3^4 is 81, 3^2 is 9. So that's 81a - 9b + 3 + 5. Combine the constants: 3 + 5 = 8. So f(3) = 81a - 9b + 8.

Now, f(-3) = a*(-3)^4 - b*(-3)^2 + (-3) + 5. Let's compute each term. (-3)^4 is also 81 because any even power of a negative is positive. (-3)^2 is 9. So the first two terms are 81a - 9b. Then, the next term is -3, and the last term is 5. So combining those: 81a - 9b - 3 + 5. The constants here are -3 +5 = 2. Therefore, f(-3) = 81a - 9b + 2. 

But we know that f(-3) = 2. So, 81a - 9b + 2 = 2. If we subtract 2 from both sides, that gives 81a - 9b = 0. 

Wait, that's interesting. So 81a - 9b = 0. Let me write that down. 

Now, looking back at f(3), which is 81a - 9b + 8. But since 81a - 9b is equal to 0, then f(3) = 0 + 8 = 8. 

Wait, so does that mean f(3) is 8? That seems too straightforward. Let me check my steps again to make sure I didn't make a mistake.

Starting with f(-3): 

a*(-3)^4 = a*81

-b*(-3)^2 = -b*9

Then, -3 (from the x term) and +5 (constant term). So f(-3) = 81a -9b -3 +5 = 81a -9b +2. That's correct. And since f(-3) = 2, then 81a -9b +2 = 2. Subtract 2: 81a -9b = 0. 

Then, f(3) is a*81 -b*9 +3 +5 = 81a -9b +8. Since 81a -9b is 0, then f(3) = 0 +8 =8. 

Hmm, seems correct. So the answer is 8. 

But let me think again. Maybe there's another way to approach this. Let's consider the function f(x). Let's split it into even and odd parts. 

An even function satisfies f(x) = f(-x), and an odd function satisfies f(x) = -f(-x). So, if we decompose f(x) into even and odd parts, the even part would be the terms with even powers of x, and the odd part would be the terms with odd powers. 

Looking at f(x) = a x⁴ - b x² + x +5. The even part is a x⁴ - b x² +5, and the odd part is x. 

So, f(x) = [a x⁴ - b x² +5] + x. 

Now, the even part is the same when x is replaced by -x, and the odd part changes sign. So, f(-x) = [a x⁴ - b x² +5] -x. 

Therefore, f(x) + f(-x) = 2*[a x⁴ - b x² +5]. 

So, if we compute f(x) + f(-x), we get twice the even part. 

But in our problem, we have f(-3) =2. Let's compute f(3) + f(-3) = 2*[a*(3)^4 - b*(3)^2 +5] = 2*[81a -9b +5]. 

But we know that f(-3) =2, so f(3) +2 = 2*(81a -9b +5). 

But from earlier, we found that 81a -9b =0. So substituting that in, the equation becomes f(3) +2 = 2*(0 +5) =10. 

Therefore, f(3) +2 =10. Subtracting 2 from both sides, f(3)=8. 

Same result. So that confirms it. 

Alternatively, if we didn't realize that, we could solve for a and b. Let's try that method as well for verification. 

We have two equations: 

From f(-3) =2: 81a -9b -3 +5 =2 => 81a -9b +2=2 => 81a -9b =0. 

But we need another equation to solve for a and b. Wait, but the problem only gives one equation. Hmm. So, perhaps we can't find unique values for a and b, but maybe we don't need them. 

Wait, but in our first method, we saw that f(3) depends on 81a -9b, which is zero. So even without knowing the exact values of a and b, we can compute f(3) as 8. 

But if we tried to solve for a and b, we would need another equation. Let's see. 

Suppose we have only one equation: 81a -9b =0. Let's write that as 9a -b =0, so b=9a. 

But without another equation, we can't find a unique solution for a and b. However, since f(3) is 81a -9b +8, and since 81a -9b is 0, it's 8. 

So, even if we don't know a and b individually, their combination 81a -9b is zero, so f(3) is 8. 

Therefore, the answer is 8. 

Just to recap, the key was recognizing that the even-powered terms (a x⁴, -b x², and the constant 5) contribute the same when x is replaced by -x, while the odd-powered term (x) changes sign. Therefore, by adding f(3) and f(-3), we eliminate the odd-powered term and can solve for f(3) using the known value of f(-3). 

Alternatively, by directly substituting x=3 and x=-3 into the function and manipulating the equations, we can find that the coefficients of the even-powered terms cancel out, leaving us with f(3) =8. 

So, after verifying through multiple methods, I'm confident that the answer is 8.
So, the value of $f(3)$ is $\boxed{8}$.
</think>