The problem asks you to evaluate and simplify $f(x + h)$ for the given function $f(x) = 5x^2 + x + 7$.

Step-by-Step Solution:

1. Start by substituting $x + h$ into the function $f(x)$: $f(x + h) = 5(x + h)^2 + (x + h) + 7$

2. Expand the quadratic term $(x + h)^2$: $(x + h)^2 = x^2 + 2xh + h^2$ So: $5(x + h)^2 = 5(x^2 + 2xh + h^2) = 5x^2 + 10xh + 5h^2$

3. Substitute this back into the function: $f(x + h) = 5x^2 + 10xh + 5h^2 + (x + h) + 7$

4. Simplify the terms: Combine all terms: $f(x + h) = 5x^2 + 10xh + 5h^2 + x + h + 7$

5. Final simplified expression: $f(x + h) = 5x^2 + 10xh + 5h^2 + x + h + 7$

Final Answer:

$f(x + h) = 5x^2 + 10xh + 5h^2 + x + h + 7$

Would you like further explanation or help with a related concept?

Related Questions:

1. How would you calculate $f(x+h) - f(x)$ for this function?
2. Can you compute $f'(x)$ using the definition of the derivative?
3. What happens when $h \to 0$ in $f(x+h) - f(x)$?
4. How would you find the vertex of the parabola $f(x) = 5x^2 + x + 7$?
5. What is the domain and range of the function $f(x) = 5x^2 + x + 7$?

Tip:

When substituting $x + h$, always expand the terms carefully, especially when squaring binomials, to avoid missing cross terms like $2xh$.