Question

which of the following is 15x² + 7x - 10 in factored form?
(5x + 10)(x - 8)
(15x - 10)(9x + 1)
none of these
(15x - 10)(x + 1)
(3x - 2)(5x + 5)

Explanation:

Step1: Recall factoring quadratic

For \(15x^2 + 7x - 12\), find \(a = 15\), \(b = 7\), \(c=-12\). Need two numbers \(m\) and \(n\) such that \(m\times n = a\times c=15\times(-12)= - 180\) and \(m + n = b = 7\).
Find \(m = 18\), \(n=-10\) (since \(18\times(-10)=-180\) and \(18+( - 10)=8\)? Wait, no, correct: \(m = 18\), \(n=-10\) no, wait \(15x^2+18x - 10x - 12\).

Step2: Factor by grouping

\(15x^2+18x - 10x - 12=3x(5x + 6)-2(5x + 6)=(3x - 2)(5x + 6)\)? Wait, no, wait original problem: Wait, maybe I misread. Wait the options: Let's check each option.
Option 1: \((x + 15)(x - 8)=x^2+7x - 120\) no.
Option 2: \((15x - 10)(8x + 1)=120x^2+15x - 80x - 10 = 120x^2 - 65x - 10\) no.
Option 4: \((15x - 10)(x + 1)=15x^2+15x - 10x - 10=15x^2 + 5x - 10\) no.
Option 5: \((3x - 2)(5x + 6)=15x^2+18x - 10x - 12=15x^2 + 8x - 12\) no. Wait, maybe the original quadratic is \(15x^2 + 7x - 12\)? Wait maybe I made a mistake. Wait let's recalculate. \(a = 15\), \(c=-12\), \(ac=-180\). Find two numbers that multiply to -180 and add to 7. 18 and -10: 18-10 = 8. No. 15 and -12: 15-12=3. No. 20 and -9: 20-9=11. No. 12 and -15: 12-15=-3. No. Wait maybe the quadratic is \(15x^2 + 8x - 12\)? Then 15x²+18x - 10x -12=3x(5x+6)-2(5x+6)=(3x - 2)(5x + 6). Wait the last option: \((3x - 2)(5x + 6)\) (assuming option 5 is \((3x - 2)(5x + 6)\)). Wait let's check the options again. Wait the options:
Wait the user's image: The last option is \((3x - 2)(5x + 6)\)? Let's check: \((3x - 2)(5x + 6)=15x^2+18x - 10x - 12=15x^2 + 8x - 12\). Wait the original problem: Maybe the original quadratic is \(15x^2 + 8x - 12\)? Or maybe I misread the coefficient. Wait the problem says "Which of the following is \(15x^2 + 7x - 12\) in factored form?" Wait no, maybe the quadratic is \(15x^2 + 8x - 12\). Wait let's check the options again. Wait the options:
Wait the first option: \((x + 15)(x - 8)=x²+7x - 120\)
Second: \((15x - 10)(8x + 1)=120x² - 65x - 10\)
Fourth: \((15x - 10)(x + 1)=15x²+5x - 10\)
Fifth: \((3x - 2)(5x + 6)=15x²+18x - 10x - 12=15x²+8x - 12\)
Wait the original problem's quadratic: Maybe a typo, but if the quadratic is \(15x^2 + 8x - 12\), then the factored form is \((3x - 2)(5x + 6)\), which is the last option. But if the quadratic is \(15x^2 + 7x - 12\), none of the options? Wait no, maybe I made a mistake. Wait let's re-express the options. Wait the third option is "none of these". Wait let's check again.
Wait the quadratic: \(15x^2 + 7x - 12\). Let's use quadratic formula: \(x=\frac{-7\pm\sqrt{49 + 720}}{30}=\frac{-7\pm\sqrt{769}}{30}\), which is not integer, so factored form with integer coefficients? Wait maybe the original quadratic is \(15x^2 + 8x - 12\), then \((3x - 2)(5x + 6)\) is correct. But if the quadratic is \(15x^2 + 7x - 12\), then none of the options. Wait the options: Let's check each:
Option 1: \((x + 15)(x - 8)=x²+7x - 120\) → degree 2, but coefficient of x² is 1, not 15.
Option 2: \((15x - 10)(8x + 1)=120x² - 65x - 10\) → x² coefficient 120.
Option 4: \((15x - 10)(x + 1)=15x²+5x - 10\) → x² coefficient 15, but linear term 5x.
Option 5: \((3x - 2)(5x + 6)=15x²+8x - 12\) → x² coefficient 15, linear term 8x.
Wait the original problem's quadratic: Maybe the user made a typo, but if the quadratic is \(15x^2 + 8x - 12\), then the last option is correct. But if it's \(15x^2 + 7x - 12\), then "none of these" (third option) is correct. Wait the third option is "none of these". Let's check the discriminant of \(15x^2 + 7x - 12\): \(D = 49 + 720 = 769\), which is not a perfect square, so it can't be factored with integer coefficients. So if the quadratic is \(15x^2 + 7x - 12\), then the answer is "none of these" (third option).

Answer:

The third option (none of these)