Question

which factors can be multiplied together to make the trinomial $5x^2 + 8x - 4$? select two options. $\square (x + 1)$ $\square (2x + 1)$ $\square (x + 2)$ $\square (5x + 1)$ $\square (5x - 2)$

Explanation:

Step1: Recall factoring trinomials

We need to factor \(5x^2 + 8x - 4\). Let's try to factor it using the box method or by finding two binomials \((ax + b)(cx + d)\) such that \(ac = 5\), \(bd=-4\), and \(ad+bc = 8\).

Step2: Test possible factors

- For option \((x + 1)\): Multiply with other candidates. \((x + 1)(5x - 4)=5x^2 - 4x+5x - 4 = 5x^2+x - 4 eq5x^2 + 8x - 4\).
- For option \((2x + 1)\): \((2x + 1)(\frac{5}{2}x - 4)\) but we need integer coefficients. Wait, let's use the box method. The middle terms in the box are \(10x\) and \(-2x\), so the trinomial is \(5x^2+10x - 2x - 4=5x(x + 2)-2(x + 2)=(5x - 2)(x + 2)\). Wait, also check \((2x + 1)\) and \((5x - 2)\)? No, wait let's check \((x + 2)\) and \((5x - 2)\): \((x + 2)(5x - 2)=5x^2-2x + 10x - 4=5x^2 + 8x - 4\). Also check \((2x + 1)\) and another? Wait, no, let's re - evaluate. Wait the options are \((x + 1)\), \((2x + 1)\), \((x + 2)\), \((5x + 1)\), \((5x - 2)\). Wait, \((2x + 1)(5x - 2)=10x^2-4x + 5x - 2=10x^2+x - 2 eq\) the trinomial. Wait, \((x + 2)(5x - 2)=5x^2-2x + 10x - 2=5x^2 + 8x - 4\). Also, wait, maybe I made a mistake. Wait the box has \(5x^2\), \(10x\), \(-2x\), \(-4\). So grouping: \(5x^2+10x-2x - 4 = 5x(x + 2)-2(x + 2)=(5x - 2)(x + 2)\). Also, let's check \((2x + 1)\) and \((5x - 2)\) no, wait \((2x + 1)(5x - 2)=10x^2 - 4x+5x - 2=10x^2+x - 2\). Wait, maybe the other way. Wait, the correct factors from the box are \((x + 2)\) and \((5x - 2)\). Let's check \((x + 2)\): \((x + 2)(5x - 2)=5x^2+8x - 4\). And \((2x + 1)\): Wait, no, maybe I messed up. Wait, let's check \((2x + 1)\) and \((5x - 2)\) again. No, \((2x + 1)(5x - 2)=10x^2 - 4x+5x - 2=10x^2+x - 2\). Wait, the correct factors are \((x + 2)\) and \((5x - 2)\). So among the options, \((x + 2)\) and \((5x - 2)\) are factors? Wait, no, the options are \((x + 1)\), \((2x + 1)\), \((x + 2)\), \((5x + 1)\), \((5x - 2)\). Wait, \((x + 2)\) and \((5x - 2)\) multiply to \(5x^2 + 8x - 4\). Also, is there another pair? Wait, maybe I made a mistake in the first calculation. Let's check \((2x + 1)\) and \((5x - 2)\) again. No, that's not it. Wait, the trinomial is \(5x^2+8x - 4\). Let's use the quadratic formula: \(x=\frac{-8\pm\sqrt{64+80}}{10}=\frac{-8\pm\sqrt{144}}{10}=\frac{-8\pm12}{10}\). So \(x=\frac{-8 + 12}{10}=\frac{4}{10}=\frac{2}{5}\) and \(x=\frac{-8-12}{10}=-2\). So the factors are \((x + 2)\) and \((5x - 2)\) (since if \(x=-2\), \(x + 2 = 0\); if \(x=\frac{2}{5}\), \(5x - 2 = 0\)). Now check the options: \((x + 2)\) is an option, and \((5x - 2)\) is an option? Wait, no, the options are \((x + 1)\), \((2x + 1)\), \((x + 2)\), \((5x + 1)\), \((5x - 2)\). Wait, also check \((2x + 1)\) and \((5x - 2)\): No, but \((x + 2)\) and \((5x - 2)\) work. Also, wait, maybe I misread the options. Wait the options are:
1. \((x + 1)\)
2. \((2x + 1)\)
3. \((x + 2)\)
4. \((5x + 1)\)
5. \((5x - 2)\)

Wait, \((x + 2)(5x - 2)=5x^2+8x - 4\), so \((x + 2)\) and \((5x - 2)\) are factors. But also, let's check \((2x + 1)\) and \((5x - 2)\): No, that's not. Wait, maybe the problem has a typo, but according to the box method, the middle terms are \(10x\) and \(-2x\), so factoring by grouping: \(5x^2+10x-2x - 4 = 5x(x + 2)-2(x + 2)=(5x - 2)(x + 2)\). So the two factors are \((x + 2)\) and \((5x - 2)\). But wait, the options also have \((2x + 1)\). Wait, no, maybe I made a mistake. Wait, let's multiply \((2x + 1)\) and \((5x - 2)\): \(10x^2-4x + 5x - 2=10x^2+x - 2 eq5x^2 + 8x - 4\). Multiply \((x + 2)\) and \((5x - 2)\): \(5x^2-2x + 10x - 4=5x^2 + 8x - 4\). Multiply \((x + 1)\) and \((5x - 2)\): \(5x^2-2x+5x - 2=5x^2 + 3x - 2 eq\). Multiply \((5x + 1)\) and \((x - 4)\): \(5x^2-20x+x - 4=5x^2-19x - 4 eq\). So the correct options are \((x + 2)\) and \((5x - 2)\). Wait, but the options given: \((x + 2)\) is option C (assuming options are labeled as A: \((x + 1)\), B: \((2x + 1)\), C: \((x + 2)\), D: \((5x + 1)\), E: \((5x - 2)\)). So the two options are C. \((x + 2)\) and E. \((5x - 2)\). Wait, but the user's options are written as: \((x + 1)\), \((2x + 1)\), \((x + 2)\), \((5x + 1)\), \((5x - 2)\). So the correct two are \((x + 2)\) and \((5x - 2)\). Wait, but let's re - check the box. The box has \(5x^2\), \(10x\), \(-2x\), \(-4\). So the trinomial is \(5x^2+10x - 2x - 4\), which factors to \((5x - 2)(x + 2)\). So the two factors are \((x + 2)\) and \((5x - 2)\).

Answer:

C. \((x + 2)\), E. \((5x - 2)\) (assuming the options are labeled as A: \((x + 1)\), B: \((2x + 1)\), C: \((x + 2)\), D: \((5x + 1)\), E: \((5x - 2)\))