Verify that
`(3x + 5y) ^(2) - 30 xy= 9x ^(2) + 25 y ^(2)`

Answer

To verify the equation \((3x + 5y)^2 - 30xy = 9x^2 + 25y^2\), we will simplify the left-hand side (LHS) and show that it equals the right-hand side (RHS).

### Step 1: Expand the LHS
We start with the left-hand side:
\[ (3x + 5y)^2 - 30xy \]
Using the formula for the square of a binomial, \((a + b)^2 = a^2 + 2ab + b^2\), where \(a = 3x\) and \(b = 5y\):
\[ (3x + 5y)^2 = (3x)^2 + 2(3x)(5y) + (5y)^2 \]
Calculating each term:
\[ (3x)^2 = 9x^2, \quad 2(3x)(5y) = 30xy, \quad (5y)^2 = 25y^2 \]
Thus, we have:
\[ (3x + 5y)^2 = 9x^2 + 30xy + 25y^2 \]

### Step 2: Substitute Back into the LHS
Now, substitute this back into the LHS:
\[ LHS = 9x^2 + 30xy + 25y^2 - 30xy \]

### Step 3: Simplify the LHS
Now, we can simplify the expression:
\[ LHS = 9x^2 + 30xy - 30xy + 25y^2 \]
The \(30xy\) terms cancel each other out:
\[ LHS = 9x^2 + 25y^2 \]

### Step 4: Compare LHS and RHS
Now, we can see that:
\[ LHS = 9x^2 + 25y^2 \]
And the right-hand side (RHS) is:
\[ RHS = 9x^2 + 25y^2 \]

Since \(LHS = RHS\), we have verified the equation.

### Conclusion
Thus, we conclude that:
\[ (3x + 5y)^2 - 30xy = 9x^2 + 25y^2 \]

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