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.