To solve for [tex]\( f(x+h) \)[/tex] for the function [tex]\( f(x) = x^2 + 5x + 1 \)[/tex], we need to substitute [tex]\( x \)[/tex] in the function with [tex]\( (x+h) \)[/tex]. Let's go through this step-by-step:
1. Start with the given function:
[tex]\[
f(x) = x^2 + 5x + 1
\][/tex]
2. Substitute [tex]\( x \)[/tex] with [tex]\( (x+h) \)[/tex] in the function:
[tex]\[
f(x+h) = (x+h)^2 + 5(x+h) + 1
\][/tex]
3. Expand [tex]\((x+h)^2\)[/tex]:
[tex]\[
(x+h)^2 = x^2 + 2xh + h^2
\][/tex]
4. Substitute back into the expression:
[tex]\[
f(x+h) = x^2 + 2xh + h^2 + 5(x+h) + 1
\][/tex]
5. Distribute the 5 in the term [tex]\( 5(x+h) \)[/tex]:
[tex]\[
5(x+h) = 5x + 5h
\][/tex]
6. Combine all the terms:
[tex]\[
f(x+h) = x^2 + 2xh + h^2 + 5x + 5h + 1
\][/tex]
7. Simplify the expression by combining like terms:
[tex]\[
f(x+h) = x^2 + 5x + 1 + 2xh + h^2 + 5h
\][/tex]
[tex]\[
f(x+h) = x^2 + 2xh + h^2 + 5x + 5h + 1
\][/tex]
Hence, the simplified expression for [tex]\( f(x+h) \)[/tex] is:
[tex]\[
f(x+h) = (h + x)^2 + 5h + 5x + 1
\][/tex]
