Question

question 5.
jackie is factoring the expression $4n^2 + 32n + 63$. both factors are in the form of $(2n + k)$.
which of the following is the greater value of $k$?
a. $21$
b. $9$
c. $7$
d. $3$

Explanation:

Step1: Set up factors

Let factors be $(2n + k_1)(2n + k_2)$. Expand: $4n^2 + 2(k_1+k_2)n + k_1k_2$.

Step2: Match coefficients

Compare with $4n^2 + 32n + 63$:
$2(k_1+k_2)=32 \Rightarrow k_1+k_2=16$;
$k_1k_2=63$.

Step3: Find k values

Solve for $k_1,k_2$: pairs of 63 factors summing to 16 are 7 and 9.

Step4: Identify greater k

Greater value is 9? No, wait—wait 7×9=63, sum 16. Wait option A is 21: if k1=21, k2=3: 21×3=63, sum=24≠16. So correct pair is 7&9, greater is 9? Wait no, wait 2(k1+k2)=32 → sum=16. 7+9=16. So greater k is 9 (option B). Wait earlier mistake: 21+3=24≠16, so 21 is invalid. So correct greater k is 9.

Wait correction:
Step3: Factors of 63: (1,63) sum=64; (3,21) sum=24; (7,9) sum=16. Only 7&9 sum to 16. So greater k is 9.

Step1: Expand assumed factors

$(2n+k_1)(2n+k_2)=4n^2+2(k_1+k_2)n+k_1k_2$.

Step2: Match coefficients

$2(k_1+k_2)=32\Rightarrow k_1+k_2=16$; $k_1k_2=63$.

Step3: Find valid k pair

Factors of 63 summing to 16: 7 and 9.

Step4: Select greater k

Greater value is 9.

Answer:

A. 21