Factor the following polynomial completely. $$ a^{2}+10 c a+9 c^{2} $$ Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $$ a^{2}+10 c a+9 c^{2}=\square $$ B. The polynomial is pirne.

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We want to factor the polynomial
$$
a^2 + 10ca + 9c^2.
$$
This expression is a quadratic in $$ a $$, where the constant term is $$ 9c^2 $$ and the coefficient of $$ a $$ is $$ 10c $$.

We look for two numbers that multiply to the constant term (ignoring the variable $$ c $$ for a moment) and add to the coefficient of $$ a $$. In this case, we need two numbers whose product is $$ 9 $$ and whose sum is $$ 10 $$. These numbers are $$ 1 $$ and $$ 9 $$.

Thus, we can write:
$$
a^2 + 10ca + 9c^2 = (a + c)(a + 9c).
$$

The factorization is complete, so the correct choice is:

A. $$ a^{2}+10 c a+9 c^{2}=(a+c)(a+9c) $$.
$$ a^{2}+10 c a+9 c^{2}=(a+c)(a+9c) $$

Factor the following polynomial completely. \[ a^{2}+10 c a+9 c^{2} \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( a^{2}+10 c a+9 c^{2}=\square \) B. The polynomial is pirne.

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