Final Answer:
The composite function f[h(x)] is equal to (8x + 5)² + (8x + 5) + 1.
Step-by-step explanation:
To evaluate the composite function f[h(x)], we substitute h(x) into f(x). The given functions are f(x) = x² + x + 1 and h(x) = 8x + 5. So, replacing x in f(x) with h(x), we get f[h(x)] = f(8x + 5). Substituting 8x + 5 into the expression for f(x), we obtain (8x + 5)² + (8x + 5) + 1.
Now, let's expand and simplify the expression.
(8x + 5)² = (8x + 5)(8x + 5) = 64x² + 40x + 40x + 25 = 64x² + 80x + 25.
Therefore, f[h(x)] = (8x + 5)² + (8x + 5) + 1 = 64x² + 80x + 25 + 8x + 5 + 1 = 64x² + 88x + 31.
In conclusion, the composite function f[h(x)] is equal to 64x² + 88x + 31. This result is obtained by substituting the expression for h(x) into the function f(x), followed by expanding and simplifying the resulting expression.