Question

which expression is equivalent to \\(n^2 + 26n + 88\\) for all values of \\(n\\)?

\\((n + 8)(n + 11)\\)
\\((n + 4)(n + 22)\\)
\\((n + 4)(n + 24)\\)
\\((n + 8)(n + 18)\\)

Explanation:

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<reasoning>

Identify the coefficients of the quadratic trinomial

The given expression is:
\[
n^2 + 26n + 88
\]
For a quadratic expression in the form \(an^2 + bn + c\), we have:
\[
a = 1,\quad b = 26,\quad c = 88
\]

Find two numbers that multiply to 88 and add to 26

We need to find two integers, \(p\) and \(q\), such that:
\[
p \cdot q = 88 \quad \text{and} \quad p + q = 26
\]
Let's list the factor pairs of 88 and find their sum:

• \(1 \times 88 \implies 1 + 88 = 89\)
• \(2 \times 44 \implies 2 + 44 = 46\)
• \(4 \times 22 \implies 4 + 22 = 26\)
• \(8 \times 11 \implies 8 + 11 = 19\)

The pair that satisfies both conditions is \(4\) and \(22\).

Write the factored form

Using the factors \(4\) and \(22\), we write the expression as:
\[
n^2 + 26n + 88 = (n + 4)(n + 22)
\]
</reasoning>

<answer>
<mcq-option>(A) \((n + 8)(n + 11)\)</mcq-option>
<mcq-correct>(B) \((n + 4)(n + 22)\)</mcq-correct>
<mcq-option>(C) \((n + 4)(n + 24)\)</mcq-option>
<mcq-option>(D) \((n + 8)(n + 18)\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the coefficients of the quadratic trinomial

The given expression is:
\[
n^2 + 26n + 88
\]
For a quadratic expression in the form \(an^2 + bn + c\), we have:
\[
a = 1,\quad b = 26,\quad c = 88
\]

Find two numbers that multiply to 88 and add to 26

We need to find two integers, \(p\) and \(q\), such that:
\[
p \cdot q = 88 \quad \text{and} \quad p + q = 26
\]
Let's list the factor pairs of 88 and find their sum:

• \(1 \times 88 \implies 1 + 88 = 89\)
• \(2 \times 44 \implies 2 + 44 = 46\)
• \(4 \times 22 \implies 4 + 22 = 26\)
• \(8 \times 11 \implies 8 + 11 = 19\)

The pair that satisfies both conditions is \(4\) and \(22\).

Write the factored form

Using the factors \(4\) and \(22\), we write the expression as:
\[
n^2 + 26n + 88 = (n + 4)(n + 22)
\]
</reasoning>

<answer>
<mcq-option>(A) \((n + 8)(n + 11)\)</mcq-option>
<mcq-correct>(B) \((n + 4)(n + 22)\)</mcq-correct>
<mcq-option>(C) \((n + 4)(n + 24)\)</mcq-option>
<mcq-option>(D) \((n + 8)(n + 18)\)</mcq-option>
</answer>

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