Factor 3x^2-4x-32 using factoring by grouping.

To factor the quadratic expression \(3x^2 - 4x - 32\) by grouping, we follow these steps:<br /><br />1. **Multiply the leading coefficient (3) by the constant term (-32):**<br /> \[<br /> 3 \times (-32) = -96<br /> \]<br /><br />2. **Find two numbers that multiply to -96 and add up to the middle coefficient (-4):**<br /> The numbers are 8 and -12 because:<br /> \[<br /> 8 \times (-12) = -96 \quad \text{and} \quad 8 + (-12) = -4<br /> \]<br /><br />3. **Rewrite the middle term (-4x) using the two numbers found:**<br /> \[<br /> 3x^2 - 4x - 32 = 3x^2 + 8x - 12x - 32<br /> \]<br /><br />4. **Group the terms in pairs:**<br /> \[<br /> (3x^2 + 8x) + (-12x - 32)<br /> \]<br /><br />5. **Factor out the greatest common factor (GCF) from each pair:**<br /> \[<br /> x(3x + 8) - 4(3x + 8)<br /> \]<br /><br />6. **Factor out the common binomial factor (3x + 8):**<br /> \[<br /> (3x + 8)(x - 4)<br /> \]<br /><br />So, the factored form of \(3x^2 - 4x - 32\) is:<br />\[<br />(3x + 8)(x - 4)<br />\]