To find \( f(x^2 - 3) \), we need to substitute \( x^2 - 3 \) into the function \( f(x) = 2x^2 + 5x - 3 \) wherever there is an \( x \).

1. Start with the original function:  
   \[
   f(x) = 2x^2 + 5x - 3
   \]

2. Substitute \( x^2 - 3 \) for \( x \) in the function:
   \[
   f(x^2 - 3) = 2(x^2 - 3)^2 + 5(x^2 - 3) - 3
   \]

3. Now, expand and simplify each term:
   - First, expand \( (x^2 - 3)^2 \):
     \[
     (x^2 - 3)^2 = x^4 - 6x^2 + 9
     \]
   - Now, multiply by 2:
     \[
     2(x^2 - 3)^2 = 2(x^4 - 6x^2 + 9) = 2x^4 - 12x^2 + 18
     \]
   - Next, expand \( 5(x^2 - 3) \):
     \[
     5(x^2 - 3) = 5x^2 - 15
     \]
   - Finally, combine all terms:
     \[
     f(x^2 - 3) = 2x^4 - 12x^2 + 18 + 5x^2 - 15 - 3
     \]

4. Simplify:
   \[
   f(x^2 - 3) = 2x^4 + (-12x^2 + 5x^2) + (18 - 15 - 3)
   \]
   \[
   f(x^2 - 3) = 2x^4 - 7x^2 + 0
   \]

Thus, the simplified expression is:
\[
f(x^2 - 3) = 2x^4 - 7x^2
\]

Would you like more details or have any questions?

Here are five related questions to expand your understanding:
1. What is the derivative of \( f(x^2 - 3) \)?
2. How would you solve \( f(x^2 - 3) = 0 \)?
3. What happens to \( f(x^2 - 3) \) as \( x \to \infty \)?
4. How do the roots of \( f(x^2 - 3) = 0 \) compare to the roots of \( f(x) = 0 \)?
5. How would you graph \( f(x^2 - 3) \)?

**Tip**: Always simplify step by step when performing substitutions, especially when dealing with polynomial expressions.

Given f(x) = 2x² + 5x - 3, find f(x² - 3)

Learn how to substitute and simplify the expression f(x² - 3) for the polynomial function f(x) = 2x² + 5x - 3. This problem covers key algebraic concepts such as function substitution and polynomial expansion. Suitable for students in Grades 9-11.

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