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.

Learn how to calculate f(2a+3) for the quadratic function f(x) = 2x^2 + 4x - 3. Follow a step-by-step solution with detailed algebraic manipulation. Suitable for high school students studying quadratic functions.

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