Answer:
-72030x⁴y +18750y⁵ +102900x³y² -52500xy⁴ +1470x²y +250y³
Explanation:
You want to simplify (7x+5y)^3–(7x–5y)^3–30y(7x+5y)(7x–5y)^3.
Substitutions
We recognize that there are some repeated expressions here that may make it easier to simplify this without multiplying it all out.
Let a = 7x+5y, b = 7x -5y. Then the expression becomes ...
a³ -b³ -30y·a·b³
= (a³ -b³) -30yb²(ab)
= (a -b)(a² +ab +b²) -30yb²·ab
Now ...
a -b = (7x +5y) -(7x -5y) = 10y
a² +b² +ab = (7x +5y)² +(7x -5y)² +(7x +5y)(7x -5y)
= (7x)² +2(7x)(5y) +(5y)² +(7x)² -2(7x)(5y) +(5y)² +(7x)² -(5y)²
= 3(7x)² +(5y)² = 147x² +25y²
So the first pair of terms expands to ...
a³ -b³ = 10y(147x² +25y²) = 1470x²y +250y³
The last term becomes (with a positive sign) ...
30yb²(ab) = 30y(7x -5y)²((7x)² -(5y)²)
= 30y((7x)² -2(7x)(5y) +(5y)²)((7x)² -(5y)²) . . . . see comment below
= 30y((7x)⁴ -(5y)⁴ -70xy((7x)² -(5y)²))
= 72030x⁴y -18750y⁵ -102900x³y² +52500xy⁴
Simplified expression
Putting it all together gives ...
= -72030x⁴y +18750y⁵ +102900x³y² -52500xy⁴ +1470x²y +250y³
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Additional comment
The difference of squares form is used several times here:
p² -q² = (p -q)(p +q)
The expression ((7x)² -2(7x)(5y) +(5y)²)((7x)² -(5y)²) is decomposed to ...
((7x)² +(5y)²)·((7x)² -(5y)²) - 2(7x)(5y)((7x)² -(5y)²)
This lets us use the difference of squares form to simplify at least part of the expression. This whole thing gets multiplied by -30y.