Question

type the correct answer in each box. use numerals instead of words. simplify the following polynomial expression. \\((5x^2 + 13x - 4) - (17x^2 + 7x - 19) + (5x - 7)(3x + 1)\\) \\(\square\\ x^2 - \square\\ x + \square\\)

Explanation:

Step1: Expand and simplify each part

First, expand \((5x - 7)(3x + 1)\):
\[ \begin{align*} (5x - 7)(3x + 1)&=5x\times3x + 5x\times1 - 7\times3x - 7\times1\\ &=15x^2 + 5x - 21x - 7\\ &=15x^2 - 16x - 7 \end{align*} \]
Then, simplify \((5x^2 + 13x - 4) - (17x^2 + 7x - 19)\):
\[ \begin{align*} &(5x^2 + 13x - 4) - (17x^2 + 7x - 19)\\ =&5x^2 + 13x - 4 - 17x^2 - 7x + 19\\ =&(5x^2 - 17x^2) + (13x - 7x) + (-4 + 19)\\ =&-12x^2 + 6x + 15 \end{align*} \]

Step2: Combine all parts

Now, add the result of the expansion and the simplified subtraction:
\[ \begin{align*} &(-12x^2 + 6x + 15) + (15x^2 - 16x - 7)\\ =&-12x^2 + 6x + 15 + 15x^2 - 16x - 7\\ =&(-12x^2 + 15x^2) + (6x - 16x) + (15 - 7)\\ =&3x^2 - 10x + 8 \end{align*} \]
Wait, there is a mistake above. Let's re - do the combination:

We have \((5x^{2}+13x - 4)-(17x^{2}+7x - 19)+(15x^{2}-16x - 7)\)

First, handle the first two polynomials:

\(5x^{2}+13x - 4-17x^{2}-7x + 19=(5x^{2}-17x^{2})+(13x - 7x)+(-4 + 19)=-12x^{2}+6x + 15\)

Then add \((15x^{2}-16x - 7)\) to \(-12x^{2}+6x + 15\):

\((-12x^{2}+6x + 15)+(15x^{2}-16x - 7)=(-12x^{2}+15x^{2})+(6x-16x)+(15 - 7)=3x^{2}-10x + 8\)? No, wait:

Wait, the original expression is \((5x^{2}+13x - 4)-(17x^{2}+7x - 19)+(5x - 7)(3x + 1)\)

We found \((5x - 7)(3x + 1)=15x^{2}-16x - 7\)

\((5x^{2}+13x - 4)-(17x^{2}+7x - 19)=5x^{2}+13x - 4-17x^{2}-7x + 19=-12x^{2}+6x + 15\)

Now add \(-12x^{2}+6x + 15\) and \(15x^{2}-16x - 7\):

\(-12x^{2}+6x + 15+15x^{2}-16x - 7=( - 12x^{2}+15x^{2})+(6x-16x)+(15 - 7)=3x^{2}-10x + 8\)? But the form is \(\square x^{2}-\square x+\square\), so we made a mistake in the sign when combining.

Wait, let's start over:

The original expression: \((5x^{2}+13x - 4)-(17x^{2}+7x - 19)+(5x - 7)(3x + 1)\)

First, expand \((5x - 7)(3x + 1)=15x^{2}+5x-21x - 7 = 15x^{2}-16x - 7\)

Then, \((5x^{2}+13x - 4)-(17x^{2}+7x - 19)=5x^{2}+13x - 4-17x^{2}-7x + 19=-12x^{2}+6x + 15\)

Now, add \(-12x^{2}+6x + 15\) and \(15x^{2}-16x - 7\):

\(-12x^{2}+6x + 15+15x^{2}-16x - 7=( - 12x^{2}+15x^{2})+(6x-16x)+(15 - 7)=3x^{2}-10x + 8\)

But the problem's final form is \(\square x^{2}-\square x+\square\), so \(3x^{2}-10x + 8\) can be written as \(3x^{2}-10x + 8\), but let's check the calculation again.

Wait, maybe I made a mistake in the sign when expanding the first subtraction. Let's re - calculate \((5x^{2}+13x - 4)-(17x^{2}+7x - 19)\):

\(5x^{2}+13x - 4-17x^{2}-7x + 19=(5 - 17)x^{2}+(13 - 7)x+(-4 + 19)=-12x^{2}+6x + 15\) (this is correct)

\((5x - 7)(3x + 1)=15x^{2}+5x-21x - 7 = 15x^{2}-16x - 7\) (this is correct)

Now add \(-12x^{2}+6x + 15\) and \(15x^{2}-16x - 7\):

\(x^{2}\) terms: \(-12x^{2}+15x^{2}=3x^{2}\)

\(x\) terms: \(6x-16x=-10x\)

constant terms: \(15 - 7 = 8\)

So the expression is \(3x^{2}-10x + 8\)

But the problem's boxed form is \(\square x^{2}-\square x+\square\), so the coefficients are 3, 10, 8.

Wait, maybe I made a mistake in the sign of the \(x\) term. Let's re - do the entire calculation:

Original expression: \((5x^{2}+13x - 4)-(17x^{2}+7x - 19)+(5x - 7)(3x + 1)\)

First, expand \((5x - 7)(3x + 1)\):

\(5x\times3x=15x^{2}\), \(5x\times1 = 5x\), \(-7\times3x=-21x\), \(-7\times1=-7\)

So \((5x - 7)(3x + 1)=15x^{2}+5x-21x - 7=15x^{2}-16x - 7\)

Then, \((5x^{2}+13x - 4)-(17x^{2}+7x - 19)=5x^{2}+13x - 4-17x^{2}-7x + 19\)

Combine like terms:

\(x^{2}\): \(5x^{2}-17x^{2}=-12x^{2}\)

\(x\): \(13x-7x = 6x\)

constants: \(-4 + 19 = 15\)

So \((5x^{2}+13x - 4)-(17x^{2}+7x - 19)=-12x^{2}+6x + 15\)

Now add \(-12x^{2}+6x + 15\) and \(15x^{2}-16x - 7\):

\(x^{2}\): \(-12x^{2}+15x^{2}=3x^{2}\)

\(x\): \(6x-16x=-10x\)

constants: \(15 - 7 = 8\)

So the simplified form is \(3x^{2}-10x + 8\), which matches the form \(\square x^{2}-\square x+\square\) with \(\square = 3\) (for \(x^{2}\)), \(\square=10\) (for \(x\)) and \(\square = 8\) (for the constant term)

Answer:

3 \(x^{2}\) - 10 \(x\) + 8, so the boxes should be filled with 3, 10, 8 respectively.