Sure! Let's find the complete factored form of the polynomial:
[tex]\[ 40 m^3 n^2 + 45 m^2 n^6 \][/tex]
### Step-by-Step Solution:
1. Identify the Greatest Common Factor (GCF):
We first look for the greatest common factor of the coefficients (40 and 45) and the variable terms.
- The GCF of the coefficients 40 and 45 is 5.
- For the variable terms, we have [tex]\( m^3 n^2 \)[/tex] and [tex]\( m^2 n^6 \)[/tex]. The common factors are [tex]\( m^2 \)[/tex] and [tex]\( n^2 \)[/tex].
Therefore, the GCF of the polynomial is [tex]\( 5m^2n^2 \)[/tex].
2. Factor out the GCF:
We factor [tex]\( 5m^2n^2 \)[/tex] out of each term in the polynomial.
[tex]\[
40m^3n^2 + 45m^2n^6 = 5m^2n^2 \cdot \left( \frac{40m^3n^2}{5m^2n^2} + \frac{45m^2n^6}{5m^2n^2} \right)
\][/tex]
3. Simplify the terms inside the parentheses:
- [tex]\[
\frac{40m^3n^2}{5m^2n^2} = \frac{40}{5} \cdot \frac{m^3}{m^2} \cdot \frac{n^2}{n^2} = 8m
\][/tex]
- [tex]\[
\frac{45m^2n^6}{5m^2n^2} = \frac{45}{5} \cdot \frac{m^2}{m^2} \cdot \frac{n^6}{n^2} = 9n^4
\][/tex]
Thus, we have:
[tex]\[
40m^3n^2 + 45m^2n^6 = 5m^2n^2 (8m + 9n^4)
\][/tex]
So, the complete factored form of the polynomial [tex]\( 40m^3n^2 + 45m^2n^6 \)[/tex] is:
[tex]\[
\boxed{5m^2n^2(8m + 9n^4)}
\][/tex]
