Certainly! Let's find the factors of the quadratic expression [tex]\(x^2 + 9x + 8\)[/tex].
1. Identify the quadratic expression: We are starting with [tex]\( x^2 + 9x + 8 \)[/tex].
2. Identify the factors of 8 that add up to 9: We need to find two numbers that multiply to 8 and add up to 9. The numbers [tex]\(1\)[/tex] and [tex]\(8\)[/tex] fit this criteria because:
[tex]\[
1 \times 8 = 8 \quad \text{and} \quad 1 + 8 = 9
\][/tex]
3. Write the expression in its factored form: We can then express the quadratic expression as:
[tex]\[
x^2 + 9x + 8 = (x + 1)(x + 8)
\][/tex]
Given this factorization, we can see how it translates to the models provided.
Analyze the models:
Model 1:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
6 & +x & + & + & + & + & + & + \\
\hline
+x & +x^2 & +x & +x & +x & +x & +x & +x \\
\hline
+ & +x & + & + & + & - & - & - \\
\hline
+ & +x & + & + & + & - & - & - \\
\hline
+ & +x & + & + & + & - & - & - \\
\hline
\end{array}
\][/tex]
Model 2:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
& +x & + & + & + & + \\
\hline
+x & +x^2 & +x & +x & +x & +x \\
\hline
+ & +x & + & + & + & + \\
\hline
+ & +x & + & + & + & + \\
\hline
\end{array}
\][/tex]
Compare with factors [tex]\((x + 1)(x + 8)\)[/tex]:
- In Model 2, looking at the structure, [tex]\(8\)[/tex] extra terms and consistent addition align with the simplicity of the expression [tex]\((x + 8)\)[/tex].
Thus, the correct model that represents the factors [tex]\((x+1)(x+8)\)[/tex] of the given quadratic expression [tex]\(x^2 + 9x + 8\)[/tex] is the second model.
