Question

\\(x^2 + 3x - 40) \div (x - 5)\\

when performing long division on numbers, divide each digit in the dividend by the divisor, then the quotient is multiplied by the divisor. subtract and bring down the next digit. then repeat this process until all the digits are divided. use the same process to divide polynomials.

set up the division as a long division problem:
\\x - 5 \overline{) x^2 + 3x - 40}\\

next, divide the first term in the dividend by the first term in the divisor. divide \\(x^2 \div x\\):
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$$\begin{array}{r} x \\\\ x - 5 \\overline{) x^2 + 3x - 40} \\end{array}$$

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multiply the quotient by the divisor. multiply \\(x \cdot (x - 5)\\):
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$$\begin{array}{r} x \\\\ x - 5 \\overline{) x^2 + 3x - 40} \\\\ \\boxed{\\phantom{x^2 - 5x}} \\end{array}$$

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Explanation:

Identify the target operation

We need to find the expression that goes into the blue box.
The prompt asks us to:
"Multiply the quotient by the divisor. Multiply \(x \cdot (x - 5)\)."

Perform the multiplication

We distribute the term \(x\) to each term inside the binomial \((x - 5)\):

$$ x \cdot (x - 5) = x \cdot x - x \cdot 5 = x^2 - 5x $$

Verify the step in long division

In polynomial long division, after placing the first term of the quotient \(x\) on top, we multiply it by the divisor \((x - 5)\) and write the result below the dividend to subtract it.
The result of this multiplication is \(x^2 - 5x\).

Answer:

Multiply the quotient by the divisor.
Multiply \(x \cdot (x - 5)\).
<blank>\(x^2 - 5x\)</blank>