To determine which given expression is equivalent to [tex]\( x^2 + 9x + 8 \)[/tex] for all values of [tex]\( x \)[/tex], we need to expand each expression and compare it to [tex]\( x^2 + 9x + 8 \)[/tex].
Let's begin with each option:
1. Option [tex]\((x+1)(x+8)\)[/tex]:
[tex]\[
(x+1)(x+8) = x \cdot x + x \cdot 8 + 1 \cdot x + 1 \cdot 8 = x^2 + 8x + x + 8 = x^2 + 9x + 8
\][/tex]
The expanded form is [tex]\( x^2 + 9x + 8 \)[/tex].
2. Option [tex]\((x+2)(x+6)\)[/tex]:
[tex]\[
(x+2)(x+6) = x \cdot x + x \cdot 6 + 2 \cdot x + 2 \cdot 6 = x^2 + 6x + 2x + 12 = x^2 + 8x + 12
\][/tex]
The expanded form is [tex]\( x^2 + 8x + 12 \)[/tex].
3. Option [tex]\((x+4)(x+4)\)[/tex]:
[tex]\[
(x+4)(x+4) = x \cdot x + x \cdot 4 + 4 \cdot x + 4 \cdot 4 = x^2 + 4x + 4x + 16 = x^2 + 8x + 16
\][/tex]
The expanded form is [tex]\( x^2 + 8x + 16 \)[/tex].
4. Option [tex]\((x+5)(x+4)\)[/tex]:
[tex]\[
(x+5)(x+4) = x \cdot x + x \cdot 4 + 5 \cdot x + 5 \cdot 4 = x^2 + 4x + 5x + 20 = x^2 + 9x + 20
\][/tex]
The expanded form is [tex]\( x^2 + 9x + 20 \)[/tex].
Now we compare each expanded form to [tex]\( x^2 + 9x + 8 \)[/tex]:
- Option [tex]\((x+1)(x+8)\)[/tex] expands to [tex]\( x^2 + 9x + 8 \)[/tex], which matches exactly.
- Option [tex]\((x+2)(x+6)\)[/tex] expands to [tex]\( x^2 + 8x + 12 \)[/tex], which does not match.
- Option [tex]\((x+4)(x+4)\)[/tex] expands to [tex]\( x^2 + 8x + 16 \)[/tex], which does not match.
- Option [tex]\((x+5)(x+4)\)[/tex] expands to [tex]\( x^2 + 9x + 20 \)[/tex], which does not match.
Thus, the expression that is equivalent to [tex]\( x^2 + 9x + 8 \)[/tex] for all values of [tex]\( x \)[/tex] is:
[tex]\[
(x+1)(x+8)
\][/tex]
