To find $f(2a+3)$ for the function $f(x) = 2x^2 + 4x - 3$, we need to substitute $2a + 3$ for $x$ in the function.

Given: $f(x) = 2x^2 + 4x - 3$

Now, we substitute $2a + 3$ in place of $x$:

$f(2a+3) = 2(2a+3)^2 + 4(2a+3) - 3$

First, let's expand $(2a + 3)^2$:

$(2a + 3)^2 = (2a + 3)(2a + 3) = 4a^2 + 12a + 9$

Now substitute this back into the function:

$f(2a+3) = 2(4a^2 + 12a + 9) + 4(2a + 3) - 3$

Distribute the 2 and the 4:

$f(2a+3) = 8a^2 + 24a + 18 + 8a + 12 - 3$

Combine like terms:

$f(2a+3) = 8a^2 + 32a + 27$

Therefore, the function $f(2a+3)$ is:

$f(2a+3) = 8a^2 + 32a + 27$

Would you like more details or have any questions?

Here are some related questions that expand on this topic:

1. How do you find the derivative of $f(x) = 2x^2 + 4x - 3$?
2. What is the vertex of the parabola given by $f(x) = 2x^2 + 4x - 3$?
3. How do you determine the roots of $f(x) = 2x^2 + 4x - 3$?
4. Can you find the integral of $f(x) = 2x^2 + 4x - 3$?
5. What is the axis of symmetry for the function $f(x) = 2x^2 + 4x - 3$?
6. How do you complete the square for the function $f(x) = 2x^2 + 4x - 3$?
7. What is the range of the function $f(x) = 2x^2 + 4x - 3$?
8. How do you graph the function $f(x) = 2x^2 + 4x - 3$?

Tip: When substituting variables, always ensure you simplify each step carefully to avoid mistakes.