\( g(x)=3 x^{2}-5 x-9 \) \[ g(x+2) \text { for the following function. } \]

To find \( g(x+2) \), you just need to substitute \( x + 2 \) into the function \( g(x) \). So, starting with \( g(x) = 3x^2 - 5x - 9 \): 1. Replace \( x \) with \( x + 2 \): \[ g(x+2) = 3(x+2)^2 - 5(x+2) - 9 \] 2. Now expand \( (x+2)^2 \): \[ (x+2)^2 = x^2 + 4x + 4 \] 3. Plug this back into the function: \[ g(x+2) = 3(x^2 + 4x + 4) - 5(x + 2) - 9 \] 4. Distribute: \[ = 3x^2 + 12x + 12 - 5x - 10 - 9 \] 5. Combine like terms: \[ g(x+2) = 3x^2 + (12x - 5x) + (12 - 10 - 9) = 3x^2 + 7x - 7 \] So, the function \( g(x+2) \) is: \[ g(x+2) = 3x^2 + 7x - 7 \] Did you know that polynomials are like the everyday mathematics of our world? They can describe everything from the trajectory of a ball to the population growth of cities! They pop up in physics, economics, and even biology. The real-world applications are as exciting as they are varied! When calculating values like \( g(x+2) \), it's easy to mix up the signs or miss a step in expanding polynomials. A good tip is to always write out each step clearly and check your work before combining terms. Misplacing a sign could turn your whole function upside down—so take it slow and double-check!