Answer:
To evaluate and fully simplify the expression f(x + h) for the given function f(x) = 5x^2 - x + 6, you need to substitute (x + h) in place of x in the function and then simplify the result.
f(x + h) = 5(x + h)^2 - (x + h) + 6
Now, let's expand and simplify this expression:
Expand (x + h)^2:
(x + h)^2 = x^2 + 2xh + h^2
Substitute this into the expression for f(x + h):
f(x + h) = 5(x^2 + 2xh + h^2) - (x + h) + 6
Distribute the 5:
f(x + h) = 5x^2 + 10xh + 5h^2 - (x + h) + 6
Now, distribute the negative sign:
f(x + h) = 5x^2 + 10xh + 5h^2 - x - h + 6
Combine like terms:
f(x + h) = 5x^2 - x + 10xh - h + 5h^2 + 6
So, the fully simplified expression for f(x + h) is:
f(x + h) = 5x^2 - x + 10xh - h + 5h^2 + 6
Explanation: