Question

1. factor $x^{2}+7x - 30$

Explanation:

Step1: Find two numbers

We need two numbers that multiply to \(-30\) (the constant term) and add up to \(7\) (the coefficient of \(x\)). Let's list the factor pairs of \(30\): \(1\) and \(30\), \(2\) and \(15\), \(3\) and \(10\), \(5\) and \(6\). Now, we need one positive and one negative number since their product is \(-30\). After checking, we find that \(10\) and \(-3\) work because \(10\times(-3)= -30\) and \(10 + (-3)=7\).

Step2: Factor the quadratic

Using the two numbers we found, we can factor the quadratic \(x^{2}+7x - 30\) as \((x + 10)(x - 3)\). We check this by expanding \((x + 10)(x - 3)=x^{2}-3x + 10x-30=x^{2}+7x - 30\), which matches the original expression.

Answer:

\((x + 10)(x - 3)\)