Question
Factor each expression. If the expression cannot be factored, indicate so \( \begin{array}{ll}\text { a. } 4 x^{2}+5 x+1 & \text { b. } x^{2}+10 x-2\end{array} \)
Ask by Wagner Ruiz. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\(4x^2+5x+1\) can be factored as \((4x + 1)(x + 1)\).
\(x^2+10x-2\) cannot be factored further.
Solution
Factor the expression \( 4x^2+5x+1 \).
Factor the expression by following steps:
- step0: Factor:
\(4x^{2}+5x+1\)
- step1: Rewrite the expression:
\(4x^{2}+\left(1+4\right)x+1\)
- step2: Calculate:
\(4x^{2}+x+4x+1\)
- step3: Rewrite the expression:
\(x\times 4x+x+4x+1\)
- step4: Factor the expression:
\(x\left(4x+1\right)+4x+1\)
- step5: Factor the expression:
\(\left(x+1\right)\left(4x+1\right)\)
The expression \(x^2+10x-2\) cannot be factored further.
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Simplify this solution The Deep Dive
To factor the expression \( 4x^{2} + 5x + 1 \), we look for two numbers that multiply to \( 4 \times 1 = 4 \) and add to \( 5 \). These numbers are \( 4 \) and \( 1 \). We can then rewrite the middle term and factor by grouping: \( 4x^2 + 4x + x + 1 = 4x(x + 1) + 1(x + 1) = (4x + 1)(x + 1) \). So, \( 4x^{2}+5x+1 \) factors to \( (4x + 1)(x + 1) \). For the expression \( x^{2}+10x-2 \), we aim to find two numbers that multiply to \( -2 \) and add to \( 10 \). Such numbers do not exist, meaning we cannot factor this expression over the integers. Therefore, \( x^{2}+10x-2 \) is not factorable in this context.
