factor completely 45r^2+150rg+125g^2?

First take out the easiest common factor: 5

45r^2 + 150rg + 125g^2
= 5 (9r^2 + 30rg + 25g^2)

This looks like the general formula (a + b)^2 = a^2 + 2ab + b^2

So keeping that in mind, we can factorise it like so:

= 5 (3r + 5g)(3r + 5g)
= 5(3r + 5g)^2

5 ( 9r^2 +30rg +25g^2)=0
( 9r^2 +15rg +15rg +25g^2 )=0
3r ( 3r +5g ) + 5g ( 3r +5g)=0
( 3r+5g)^2=0
3r+5g=0, 3r=5g, r/g= 5/3.

45r^2 + 150rg + 125g^2
= 5(9r^2 + 30rg + 25g^2)
= 5(9r^2 + 15rg + 15rg + 25g^2)
= 5(3r + 5g)(3r + 5g)
= 5(3r + 5g)^2

5 ( 9 r ² + 30 r g + 25 g ² )
5 ( 3 r + 5 g ) ( 3 r + 5 g )
5 ( 3 r + 5 g ) ²