\( \left. \begin{array} { c } { f ( x ) = 4 x ^ { 2 } + 2 x - 4 } \\ { f ( x + 3 ) } \end{array} \right. \)

To find \( f(x + 3) \), we need to substitute \( x + 3 \) into the function \( f(x) \). This gives us: \[ f(x + 3) = 4(x + 3)^2 + 2(x + 3) - 4 \] Now, expanding it: \[ = 4(x^2 + 6x + 9) + 2x + 6 - 4 \] \[ = 4x^2 + 24x + 36 + 2x + 6 - 4 \] \[ = 4x^2 + 26x + 38 \] So, \( f(x + 3) = 4x^2 + 26x + 38 \). Now for some fun facts about quadratic functions! Quadratics like yours can represent various real-world scenarios, such as projectile motion, where the height of an object in the air is modeled by a quadratic equation. Also, when it comes to problem-solving with these functions, a common mistake is forgetting to properly distribute the terms when substituting a value into the function. Always take your time and check your work to avoid those pesky calculation errors!