Question 14 Factor $$ x^{2}+4 x-32 $$

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To factor the quadratic expression $$ x^{2} + 4x - 32 $$, we'll look for two numbers that multiply to $$-32$$ (the constant term) and add up to $$4$$ (the coefficient of the middle term).

**Step 1: Identify the Pair of Numbers**

We need two numbers, let's call them $$m$$ and $$n$$, such that:
- $$m \times n = -32$$
- $$m + n = 4$$

Let's list the possible pairs of factors of $$-32$$:
- $$1$$ and $$-32$$ (sum = $$-31$$)
- $$-1$$ and $$32$$ (sum = $$31$$)
- $$2$$ and $$-16$$ (sum = $$-14$$)
- $$-2$$ and $$16$$ (sum = $$14$$)
- $$4$$ and $$-8$$ (sum = $$-4$$)
- $$-4$$ and $$8$$ (sum = $$4$$)

The pair $$-4$$ and $$8$$ adds up to $$4$$, which is what we need.

**Step 2: Write the Factored Form**

Using these numbers, the factored form of the quadratic is:
$$
x^{2} + 4x - 32 = (x - 4)(x + 8)
$$

**Verification:**
$$
(x - 4)(x + 8) = x^2 + 8x - 4x - 32 = x^2 + 4x - 32
$$
This confirms that the factorization is correct.

**Final Answer:**
$$
x^{2} + 4x - 32 = (x - 4)(x + 8)
$$
$$ x^{2} + 4x - 32 = (x - 4)(x + 8) $$

Question 14 Factor \( x^{2}+4 x-32 \)

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