What must be subtracted from `19x^(4) + 2x^(3) + 30x - 37` to get `8x^(4) + 22x^(3) - 7x - 60` ?

Answer

To solve the problem of what must be subtracted from \(19x^4 + 2x^3 + 30x - 37\) to get \(8x^4 + 22x^3 - 7x - 60\), we can follow these steps:

### Step 1: Identify the expressions
Let:
- \(A = 19x^4 + 2x^3 + 30x - 37\)
- \(C = 8x^4 + 22x^3 - 7x - 60\)

We need to find \(B\) such that:
\[ A - B = C \]

### Step 2: Rearranging the equation
From the equation \(A - B = C\), we can rearrange it to find \(B\):
\[ B = A - C \]

### Step 3: Substitute the expressions for \(A\) and \(C\)
Now, substitute the values of \(A\) and \(C\):
\[ B = (19x^4 + 2x^3 + 30x - 37) - (8x^4 + 22x^3 - 7x - 60) \]

### Step 4: Distributing the negative sign
Distributing the negative sign to the terms in \(C\):
\[ B = 19x^4 + 2x^3 + 30x - 37 - 8x^4 - 22x^3 + 7x + 60 \]

### Step 5: Combine like terms
Now, we will combine the like terms:
- For \(x^4\) terms: \(19x^4 - 8x^4 = 11x^4\)
- For \(x^3\) terms: \(2x^3 - 22x^3 = -20x^3\)
- For \(x\) terms: \(30x + 7x = 37x\)
- For constant terms: \(-37 + 60 = 23\)

Putting it all together:
\[ B = 11x^4 - 20x^3 + 37x + 23 \]

### Final Answer
Thus, the expression that must be subtracted is:
\[ B = 11x^4 - 20x^3 + 37x + 23 \]

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