To determine which expression is equivalent to [tex]\( x^2 + 9x + 8 \)[/tex] for all values of [tex]\( x \)[/tex], we can factorize the quadratic expression and compare it to the given options.
Here’s the step-by-step process of checking each option:
1. Expanding each given expression:
- Option 1: [tex]\((x+1)(x+8)\)[/tex]
[tex]\[
(x+1)(x+8) = x(x+8) + 1(x+8) = x^2 + 8x + x + 8 = x^2 + 9x + 8
\][/tex]
This expands to [tex]\( x^2 + 9x + 8 \)[/tex], which matches the original expression. Hence, this option is correct.
- Option 2: [tex]\((x+2)(x+6)\)[/tex]
[tex]\[
(x+2)(x+6) = x(x+6) + 2(x+6) = x^2 + 6x + 2x + 12 = x^2 + 8x + 12
\][/tex]
This expands to [tex]\( x^2 + 8x + 12 \)[/tex], which does not match the original expression.
- Option 3: [tex]\((x+4)(x+4)\)[/tex]
[tex]\[
(x+4)(x+4) = x(x+4) + 4(x+4) = x^2 + 4x + 4x + 16 = x^2 + 8x + 16
\][/tex]
This expands to [tex]\( x^2 + 8x + 16 \)[/tex], which does not match the original expression.
- Option 4: [tex]\((x+5)(x+4)\)[/tex]
[tex]\[
(x+5)(x+4) = x(x+4) + 5(x+4) = x^2 + 4x + 5x + 20 = x^2 + 9x + 20
\][/tex]
This expands to [tex]\( x^2 + 9x + 20 \)[/tex], which does not match the original expression.
After verifying each option, we find that:
- Option 1: [tex]\((x+1)(x+8)\)[/tex] is the only choice that correctly factors the expression [tex]\( x^2 + 9x + 8 \)[/tex].
Therefore, 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].
