Question

1. $3x^2 - 11x - 4$

Explanation:

Step1: Find two numbers

We need two numbers that multiply to \(3\times(-4)= -12\) and add up to \(-11\). The numbers are \(-12\) and \(1\) since \(-12\times1 = -12\) and \(-12 + 1 = -11\).

Step2: Split the middle term

Rewrite the quadratic as \(3x^{2}-12x + x - 4\).

Step3: Group and factor

Group the first two and last two terms: \((3x^{2}-12x)+(x - 4)\). Factor out \(3x\) from the first group: \(3x(x - 4)+1(x - 4)\). Now factor out \((x - 4)\): \((3x + 1)(x - 4)\).
For the box method, we split the middle term into \(-12x\) and \(x\). So the top - left box is \(3x^{2}\), top - right box is \(x\), bottom - left box is \(-12x\), bottom - right box is \(-4\). Then we factor rows and columns:

• Row 1: \(x(3x + 1)\)
• Row 2: \(-4(3x + 1)\)
• Column 1: \(3x(x - 4)\)
• Column 2: \(1(x - 4)\)

Answer:

\((3x + 1)(x - 4)\)