Q. Which expression is a factor of \(x^2+7x-30\)?

State the problem and goal:

Factor the quadratic expression \(x^{2} + 7x – 30\) into a product of linear factors.

Identify coefficients for a standard quadratic \(ax^{2}+bx+c\):

Here \(a=1\), \(b=7\), \(c=-30\).

Find two numbers whose product equals \(a\cdot c\) and whose sum equals \(b\):

We need integers \(m\) and \(n\) such that

\(m\cdot n = a\cdot c = 1\cdot(-30) = -30\)

and

\(m + n = b = 7\).

List factor pairs of \(-30\) and their sums to find \(m\) and \(n\):

• \((-1, 30)\) gives sum \(-1 + 30 = 29\)
• \((1, -30)\) gives sum \(1 + (-30) = -29\)
• \((-2, 15)\) gives sum \(-2 + 15 = 13\)
• \((2, -15)\) gives sum \(2 + (-15) = -13\)
• \((-3, 10)\) gives sum \(-3 + 10 = 7\)
• \((3, -10)\) gives sum \(3 + (-10) = -7\)

The pair that gives the required sum \(7\) is \(m=-3\) and \(n=10\).

Use these numbers to split the middle term and factor by grouping:

Write

\(x^{2} + 7x – 30 = x^{2} – 3x + 10x – 30\).

Group terms:

\((x^{2} – 3x) + (10x – 30)\).

Factor each group:

\(x(x – 3) + 10(x – 3)\).

Factor out the common binomial factor \((x – 3)\):

\((x – 3)(x + 10)\).

Verify by expanding (optional check):

\((x – 3)(x + 10) = x^{2} + 10x – 3x – 30 = x^{2} + 7x – 30\).

Conclusion — the linear factors are

\(x – 3\) and \(x + 10\).

Therefore, an expression that is a factor of \(x^{2} + 7x – 30\) is \(x – 3\). Another factor is \(x + 10\).