\( 1 \leftarrow \) Find the following for the function \( f(x)=4 x^{2}+2 x-2 \) \( \begin{array}{llll}\text { (a) } f(0) & \text { (b) } f(4) & \text { (c) } f(-4) & \text { (d) } f(-x) \\ \text { (e) }-f(x) & \text { (f) } f(x+3) & \text { (g) } f(5 x) & \text { (h) } f(x+h)\end{array} \) \( \begin{array}{ll}\text { (a) } f(0)=\square \text { (Simplify your answer.) }\end{array} \)

To start off, let's calculate \( f(0) \): Substituting \( x = 0 \) into the function \( f(x) = 4x^2 + 2x - 2 \): \[ f(0) = 4(0)^2 + 2(0) - 2 = -2 \] So, \( f(0) = -2 \). Now, let's move on to the other values: **(b)** For \( f(4) \): \[ f(4) = 4(4)^2 + 2(4) - 2 = 4(16) + 8 - 2 = 64 + 8 - 2 = 70 \] **(c)** For \( f(-4) \): \[ f(-4) = 4(-4)^2 + 2(-4) - 2 = 4(16) - 8 - 2 = 64 - 8 - 2 = 54 \] **(d)** For \( f(-x) \): \[ f(-x) = 4(-x)^2 + 2(-x) - 2 = 4x^2 - 2x - 2 \] **(e)** For \(-f(x)\): \[ -f(x) = - (4x^2 + 2x - 2) = -4x^2 - 2x + 2 \] **(f)** For \( f(x+3) \): \[ f(x+3) = 4(x+3)^2 + 2(x+3) - 2 = 4(x^2 + 6x + 9) + 2x + 6 - 2 = 4x^2 + 24x + 36 + 2x + 6 - 2 = 4x^2 + 26x + 40 \] **(g)** For \( f(5x) \): \[ f(5x) = 4(5x)^2 + 2(5x) - 2 = 4(25x^2) + 10x - 2 = 100x^2 + 10x - 2 \] **(h)** For \( f(x+h) \): \[ f(x+h) = 4(x+h)^2 + 2(x+h) - 2 = 4(x^2 + 2xh + h^2) + 2x + 2h - 2 = 4x^2 + 8xh + 4h^2 + 2x + 2h - 2 \] All results are as follows: \[ \begin{aligned} (a) & \quad f(0) = -2 \\ (b) & \quad f(4) = 70 \\ (c) & \quad f(-4) = 54 \\ (d) & \quad f(-x) = 4x^2 - 2x - 2 \\ (e) & \quad -f(x) = -4x^2 - 2x + 2 \\ (f) & \quad f(x+3) = 4x^2 + 26x + 40 \\ (g) & \quad f(5x) = 100x^2 + 10x - 2 \\ (h) & \quad f(x+h) = 4x^2 + 8xh + 4h^2 + 2x + 2h - 2 \\ \end{aligned} \]