<Question>
If $f(x)=5x^2+3x+4$, what is the value of $f(-2)$?
</Question>

<think>
<Plan_1>
Okay, so I need to find the value of the function f(x) when x is -2. The function is given as f(x) = 5x² + 3x + 4. Hmm, let me think about how to approach this. 

First, I remember that evaluating a function at a specific point just means substituting the variable x with that value. So in this case, everywhere there's an x in the equation, I should replace it with -2 and then compute the result. Let me write that down step by step to make sure I don't make any mistakes.

<prompt_1.1> Substitute \( x = -2 \) into the function \( f(x) = 5x^2 + 3x + 4 \) and begin simplifying each term. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
Starting with the original function: f(x) = 5x² + 3x + 4. Substituting x = -2, we get:

f(-2) = 5*(-2)² + 3*(-2) + 4.

Alright, let's break this down term by term. The first term is 5 times (-2) squared. I need to calculate (-2) squared first. Remembering that when you square a negative number, the result is positive because a negative times a negative is a positive. So (-2)² is (-2)*(-2) which equals 4. Then multiply that by 5: 5*4 = 20. So the first term is 20.

Moving on to the second term: 3 times (-2). That's straightforward multiplication. 3*(-2) = -6. So the second term is -6.

The third term is just a constant, 4. So that stays as it is.

Now, putting all the terms back together: 20 + (-6) + 4. Let's compute that step by step. 

Starting with 20 and -6: 20 + (-6) is the same as 20 - 6, which equals 14. Then add the last term, 4, to that result: 14 + 4 = 18.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Wait, let me check that again to be sure. Sometimes signs can trip me up, especially with negative numbers. Let me verify each step.

<prompt_2.1> Verify each step of the calculation for \( f(-2) = 5(-2)^2 + 3(-2) + 4 \). </prompt_2.1>

Wait a second, let me try another way just to confirm. Maybe using a different order of operations. Let's compute all the terms first and then add them up.

<prompt_2.2> Recalculate \( f(-2) \) using a different order of operations to confirm the result. </prompt_2.2>

Another way to check is to use the order of operations (PEMDAS): Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. Let's apply that.

<prompt_2.3> Apply the order of operations (PEMDAS) to verify the calculation of \( f(-2) \). </prompt_2.3>
</Plan_2>

<Execution_2>
<execution_2.1>
First term: (-2) squared is 4, multiplied by 5 gives 20. Correct. Second term: 3*(-2) is indeed -6. Third term: 4. Adding them: 20 - 6 + 4. 

Alternatively, adding them all at once: 20 - 6 is 14, then 14 + 4 is 18. So f(-2) should be 18.
</execution_2.1>

<execution_2.2>
First term: 5*(-2)². Let's calculate (-2)² first, which is 4, then multiply by 5: 5*4=20. Correct. 

Second term: 3*(-2) = -6. Correct. 

Third term: 4. Correct. 

Adding them: 20 + (-6) = 14, then 14 + 4 = 18.
</execution_2.2>

<execution_2.3>
Starting with the exponent: (-2)² is 4. Then multiply by 5: 5*4=20. Then 3*(-2)=-6. The constant is 4. So adding them up: 20 -6 +4. 

20 -6 is 14, then 14 +4 is 18.
</execution_2.3>
</Execution_2>

<Plan_3>
Based on execution_2.1, execution_2.2, execution_2.3:
Wait, but just to be thorough, let me compute it once more. Maybe even expand the terms again.

<prompt_3.1> Re-verify the calculation of \( f(-2) \) by expanding the terms again and ensuring the correct order of operations is followed. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
5*(-2)^2 = 5*(4) = 20. 

3*(-2) = -6. 

Plus 4. 

20 -6 +4. 

20 -6 is 14. 14 +4 is 18. 

No mistakes here. 

Alternatively, if I miscalculated (-2)^2 as -4, that would be wrong, but no, exponents come before multiplication, so (-2)^2 is definitely positive 4. So 5*4 is 20, not negative. 

Another possible mistake could be in the sign when multiplying 3 and -2. 3 times -2 is -6, right. So adding 20, subtracting 6, and adding 4. 

20 -6 is 14, 14 +4 is 18. 

I think I've verified this enough times. Each step checks out. So the final answer should be 18.
</execution_3.1>
</Execution_3>

<Final_answer>
Therefore, the value of $f(-2)$ is $\boxed{18}$.
</Final_answer>
</think>