If f(x) = x² + 4x then find f(0)+ f(2) 
              x  + 6                    f(2) + f(3) 

We are given the following functions and need to find the value of the given expression:

\[
f(x) = x^2 + 4x
\]

We are tasked with calculating:

\[
\frac{f(0) + f(2)}{f(2) + f(3)}
\]

### Step 1: Find \( f(0) \)
\[
f(0) = 0^2 + 4(0) = 0
\]

### Step 2: Find \( f(2) \)
\[
f(2) = 2^2 + 4(2) = 4 + 8 = 12
\]

### Step 3: Find \( f(3) \)
\[
f(3) = 3^2 + 4(3) = 9 + 12 = 21
\]

### Step 4: Substitute the values into the expression

\[
\frac{f(0) + f(2)}{f(2) + f(3)} = \frac{0 + 12}{12 + 21} = \frac{12}{33}
\]

### Step 5: Simplify the fraction
\[
\frac{12}{33} = \frac{4}{11}
\]

Thus, the value of the expression is:

\[
\boxed{\frac{4}{11}}
\]

Would you like more details on any part of this solution or have further questions?

Here are 5 related questions:
1. How would you solve this if \( f(x) \) were a cubic function instead of a quadratic?
2. What happens if you change the denominator to \( f(1) + f(2) \)?
3. Can you calculate the derivative of \( f(x) \) and find critical points?
4. How do you evaluate \( f(x) \) for negative values of \( x \)?
5. What if you add a constant term to \( f(x) \), like \( f(x) = x^2 + 4x + 5 \)?

**Tip:** Always check if you can simplify a fraction by finding the greatest common divisor of the numerator and denominator.

If f(x) = x² + 4x then find f(0) + f(2) / (f(2) + f(3))

Learn how to evaluate the rational expression (f(0) + f(2)) / (f(2) + f(3)) where f(x) = x² + 4x. This step-by-step solution covers function evaluation, simplification, and core algebraic concepts. Suitable for students in Grades 8-10.

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