find the difference quotient \\( \\frac { f ( x + h ) - f ( x ) } { h } \\), where \\( h \\neq 0 \\), for…

find the difference quotient \\( \\frac { f ( x + h ) - f ( x ) } { h } \\), where \\( h \\neq 0 \\), for the function below.\n\n\\( f ( x ) = - 4 x ^ { 2 } - 6 x + 7 \\)\n\nsimplify your answer as much as possible.\n\n\\( \\frac { f ( x + h ) - f ( x ) } { h } = \\square \\)

find the difference quotient \\( \\frac { f ( x + h ) - f ( x ) } { h } \\), where \\( h \\neq 0 \\), for the function below.\n\n\\( f ( x ) = - 4 x ^ { 2 } - 6 x + 7 \\)\n\nsimplify your answer as much as possible.\n\n\\( \\frac { f ( x + h ) - f ( x ) } { h } = \\square \\)

Answer

Explanation:

Step1: Find (f(x + h))

Given (f(x)=-4x^{2}-6x + 7), then (f(x + h)=-4(x + h)^{2}-6(x + h)+7). Using ((a + b)^{2}=a^{2}+2ab + b^{2}), we have (f(x + h)=-4(x^{2}+2xh+h^{2})-6x-6h + 7=-4x^{2}-8xh-4h^{2}-6x-6h + 7).

Step2: Calculate (f(x + h)-f(x))

(f(x + h)-f(x)=(-4x^{2}-8xh-4h^{2}-6x-6h + 7)-(-4x^{2}-6x + 7)) (=-4x^{2}-8xh-4h^{2}-6x-6h + 7 + 4x^{2}+6x - 7) (=-8xh-4h^{2}-6h).

Step3: Find (\frac{f(x + h)-f(x)}{h})

(\frac{f(x + h)-f(x)}{h}=\frac{-8xh-4h^{2}-6h}{h}), since (h\neq0), we can factor out (h) from the numerator: (\frac{h(-8x - 4h-6)}{h}=-8x-4h - 6).

Answer:

(-8x-4h - 6)