How do you factor `9x^2+24xy+16y^2`?

Note that:

`9x^2 = (3x)^2`

`24xy = 2(3x)(4y)`

`16y^2 = (4y)^2`

So:

`9x^2+24xy+16y^2 = (3x)^2+2(3x)(4y)+(4y)^2`

is in the form:

`A^2+2AB+B^2 = (A+B)^2`

So putting `A=3x` and `B=4y` we have:

`9x^2+24xy+16y^2 = (3x+4y)^2`

We are given:

`9x^2 + 24xy + 16y^2`

We want to obtain an expression of the form:

`(ax+by)(cx+dy)`

where `a,b,c,d` are integers (not necessarily unique from each other).

Expanding this form we get:

`acx^2 + adxy +bcxy + bdy^2`

`acx^2 +(ab+cd)xy + bdy^2`

From the expression we are given we must satisfy the following equations:

`ac = 9`
`bd = 16`
`ab+cd = 24`

For `(a,c)` we can have (eliminating any repeats):
`(1,9),(3,3),cancel{(9,1)}`

For `(b,d)` we can have (eliminating any repeats):
`(1,16),(2,8),(4,4),cancel{(8,2)},cancel{(16,1)}`

With these options, we now need to find the ones that when combined will give us the `ab+cd = 24`:

`(1,9)(1,16) -> 1xx1+9xx16=145`
`(1,9)(2,8) -> 1xx2+9xx8 =74`
`(1,9)(4,4) -> 1xx4+9xx4 = 40`
`(3,3)(1,16) -> 3xx1+3xx16= 51`
`(3,3)(2,8) -> 3xx2+3xx8=30`
`color(blue)((3,3)(4,4) -> 3xx4+3xx4 =24)`

Hence, we must choose `(a,c) = (3,3)` and `(b,d) = (4,4)`

So the factored expression `(ax+by)(cx+dy)` is:

`9x^2 + 24xy + 16y^2= color(green){(3x+4y)(3x+4y)}`