Question

1. $4x^2 - 31x + 21$

Explanation:

Step1: Find two numbers

We need two numbers that multiply to \(4\times21 = 84\) and add up to \(-31\). The numbers are \(-28\) and \(-3\) since \((-28)\times(-3)=84\) and \((-28)+(-3)=-31\).

Step2: Rewrite the middle term

Rewrite \(-31x\) as \(-28x - 3x\) in the quadratic expression:
\(4x^{2}-28x - 3x + 21\)

Step3: Factor by grouping

Group the first two terms and the last two terms:
\((4x^{2}-28x)+(-3x + 21)\)
Factor out the greatest common factor from each group:
\(4x(x - 7)-3(x - 7)\)

Step4: Factor out the common binomial factor

Factor out \((x - 7)\) from the expression:
\((4x - 3)(x - 7)\)

Answer:

\((4x - 3)(x - 7)\)