Question

factor $9n^2 + 90n + 63$.
a. $9(n+9)(n+7)$
b. $9(n+10)(n+6)$
c. $9(n^2 + 10n + 6)$
d. $9(n^2 + 10n + 7)$

Explanation:

Step1: Factor out GCF

Identify greatest common factor (GCF) of $9n^2$, $90n$, $63$ is 9.
$\frac{9n^2}{9} + \frac{90n}{9} + \frac{63}{9} = n^2 + 10n + 7$

Step2: Verify factored form

Multiply 9 with the simplified quadratic:
$9(n^2 + 10n + 7)$

Step3: Check quadratic factorability

Check if $n^2+10n+7$ can be factored: discriminant $=10^2-4(1)(7)=100-28=72$, which is not a perfect square, so it cannot be factored into integer binomials.

Answer:

d. $9(n^{2}+10n+7)$