Question

1. $6x^2 + 41x + 63$

Explanation:

Step1: Find two numbers that multiply to \(6\times63 = 378\) and add up to 41.

We need to factor 378 to find such numbers. The factors of 378 are: 1 & 378, 2 & 189, 3 & 126, 6 & 63, 7 & 54, 9 & 42, 14 & 27, 18 & 21. Among these, 14 and 27 add up to 41 (\(14 + 27=41\)) and multiply to \(14\times27 = 378\).

Step2: Rewrite the middle term using these two numbers.

We can rewrite the quadratic expression \(6x^{2}+41x + 63\) as \(6x^{2}+14x+27x + 63\).

Step3: Group the terms and factor out the common factors.

Group the first two terms and the last two terms: \((6x^{2}+14x)+(27x + 63)\).
Factor out the greatest common factor (GCF) from each group. From the first group, the GCF of \(6x^{2}\) and \(14x\) is \(2x\), so we get \(2x(3x + 7)\). From the second group, the GCF of \(27x\) and \(63\) is 9, so we get \(9(3x + 7)\).

Step4: Factor out the common binomial factor.

Now we have \(2x(3x + 7)+9(3x + 7)\). The common binomial factor is \((3x + 7)\), so we factor that out: \((2x + 9)(3x + 7)\).

Answer:

\((2x + 9)(3x + 7)\)