Question

Factor the expression 32^2-60ab+25b^2 into a product of binomials

Answer 1

Lots of factor pairs but some good guessing will simplify:
you're looking for something like (Ax+By)(Cx+Dy)
multiplied out that gives ACx^2+(BC+AD)xy+BDy^2
Let's guess B and D are 5
(Ax+5y)(Cx+5)
Now we need AC=32 and 5A+5C=60
Factors of 32: 1,2,4,8,16,32
5x4=20, 5x8=40 giving our 60 added together
Final result going back to your original a's and b's:
(4a-5b)(8a-5b)

Answer 2

Maybe you'll like this better:
\[32a^2-60ab+25b^2\]
Look to separate the middle term into two parts so that the first has a common factor with 32 and the second a common factor with 25:
\[32a^2-40ab-20ab+25b^2\]
Now pull those common factors out:
\[8a(4a-5b)-5b(4a-5b)\]
Note that the bits in the parentheses are the same; this is critical or you can't do the next step which is to pull that part out:
(4a-5b)(8a-5b)