Factor the trinomial x2 + x - 12. Identify the values of b and c, and find two integers r and s such that their product is c and their sum is b.

Here are the solutions for the factorization of $x^2 + x - 12$.

3.2.1 What is the value of b and c in the trinomial?

The standard form of a trinomial is $ax^2 + bx + c$.
Comparing $x^2 + x - 12$ with $ax^2 + bx + c$:
$b = 1$
$c = -12$

3.2.2 Use the table below to determine the two integers, r and s.

We need two integers whose product is $c = -12$ and whose sum is $b = 1$.

Factors of -12	Product of the two factors	Sum of the two factors	Result
$1, -12$	$1 \times (-12) = -12$	$1 + (-12) = -11$
$-1, 12$	$-1 \times 12 = -12$	$-1 + 12 = 11$
$2, -6$	$2 \times (-6) = -12$	$2 + (-6) = -4$
$-2, 6$	$-2 \times 6 = -12$	$-2 + 6 = 4$
$3, -4$	$3 \times (-4) = -12$	$3 + (-4) = -1$
$\mathbf{-3, 4}$	$\mathbf{-3 \times 4 = -12}$	$\mathbf{-3 + 4 = 1}$	Correct

The two integers are $\mathbf{-3}$ and $\mathbf{4}$.

3.2.3 In the table below, the written explanation of the steps have been provided, show the mathematical steps for the explanations given.

Mathematical steps	Explanation
$x^2 + 4x - 3x - 12$	Rewrite the middle term of the trinomial using the values from the chart above.
$(x^2 + 4x) - (3x + 12)$	Group pairs of terms.
$x(x + 4)$	Factor out the HCF of the first group.
$-3(x + 4)$	Factor out the HCF of the second group.
$(x + 4)(x - 3)$	Factor out HCF of the two terms.
$(x + 4)(x - 3)$	The final factorized answer.