Question

1. $x^2 + 3x - 54$

Explanation:

Step1: Find two numbers

We need two numbers that multiply to -54 and add up to 3. Let's list the factor pairs of 54: (1, 54), (2, 27), (3, 18), (6, 9). Since the product is negative, one number is positive and the other is negative. We find that 9 and -6 work because \(9\times(-6)= -54\) and \(9 + (-6)=3\).

Step2: Rewrite the middle term

Rewrite the quadratic expression \(x^{2}+3x - 54\) by splitting the middle term using the two numbers we found: \(x^{2}+9x - 6x - 54\).

Step3: Group and factor

Group the first two terms and the last two terms: \((x^{2}+9x)+(-6x - 54)\). Factor out the greatest common factor from each group: \(x(x + 9)-6(x + 9)\).

Step4: Factor out the common binomial

Now, factor out the common binomial factor \((x + 9)\): \((x + 9)(x - 6)\).

Answer:

\((x + 9)(x - 6)\)