Question

1. $k^2 - 12k + 27$
2. $n^2 + n + 2$
3. $r^2 + 3r - 28$
4. $m^2 + 4m - 45$

Explanation:

Step1: Factor $k^2-12k+27$

Find two numbers that multiply to $27$ and add to $-12$: $-3$ and $-9$.
$\boldsymbol{k^2-12k+27=(k-3)(k-9)}$

Step2: Check if $n^2+n+2$ factors

No real number pair multiplies to $2$ and adds to $1$.
$\boldsymbol{n^2+n+2}$ (prime, cannot be factored over reals)

Step3: Factor $r^2+3r-28$

Find two numbers that multiply to $-28$ and add to $3$: $7$ and $-4$.
$\boldsymbol{r^2+3r-28=(r+7)(r-4)}$

Step4: Factor $m^2+4m-45$

Find two numbers that multiply to $-45$ and add to $4$: $9$ and $-5$.
$\boldsymbol{m^2+4m-45=(m+9)(m-5)}$

Answer:

1. $\boldsymbol{(k-3)(k-9)}$
2. $\boldsymbol{n^2+n+2}$ (cannot be factored over real numbers)
3. $\boldsymbol{(r+7)(r-4)}$
4. $\boldsymbol{(m+9)(m-5)}$