Question

what are the factors of $6x^2 + 37x - 60$?

• $3x - 4$ and $2x + 15$
• $3x + 4$ and $2x - 15$
• $2(x - 2)$ and $3(x + 5)$
• $2(x + 2)$ and $3(x - 5)$

Explanation:

Step1: Recall factoring quadratic trinomial \(ax^2 + bx + c\) (here \(a = 6\), \(b = 37\), \(c=-60\)). We need two numbers \(m\) and \(n\) such that \(m\times n=a\times c = 6\times(-60)=- 360\) and \(m + n=b = 37\).

Find \(m\) and \(n\): The numbers are \(45\) and \(- 8\) since \(45\times(-8)=-360\) and \(45+( - 8)=37\).

Step2: Rewrite the middle term using \(m\) and \(n\):

\(6x^{2}+45x - 8x-60\)

Step3: Group the terms:

\((6x^{2}+45x)-(8x + 60)\)

Step4: Factor out the GCF from each group:

\(3x(2x + 15)-4(2x + 15)\)

Step5: Factor out the common binomial factor \((2x + 15)\):

\((3x - 4)(2x + 15)\)
We can also check by multiplying the options:

• For \(3x - 4\) and \(2x + 15\): \((3x-4)(2x + 15)=3x\times2x+3x\times15-4\times2x-4\times15 = 6x^{2}+45x-8x - 60=6x^{2}+37x - 60\) (matches the given quadratic).
• For \(3x + 4\) and \(2x-15\): \((3x + 4)(2x-15)=6x^{2}-45x + 8x-60=6x^{2}-37x - 60\) (does not match).
• For \(2(x - 2)\) and \(3(x + 5)\): \(2(x - 2)\times3(x + 5)=6(x - 2)(x + 5)=6(x^{2}+5x-2x - 10)=6x^{2}+18x-60\) (does not match).
• For \(2(x + 2)\) and \(3(x - 5)\): \(2(x + 2)\times3(x - 5)=6(x + 2)(x - 5)=6(x^{2}-5x + 2x-10)=6x^{2}-18x - 60\) (does not match).

Answer:

A. \(3x - 4\) and \(2x + 15\)