The first step is to square each side of the inequality.

\(\displaystyle \left( \sqrt{x + 19} \right)^2> \left( x + 1 \right)^2\)

Now simplify each side.

\(\displaystyle x + 19 > x^{2} + 2 x + 1\)

Now subtract the left side of the inequality to make it zero, so that we can use the quadratic formula.

\(\displaystyle x + 19 -\left( x + 19 \right)> x^{2} + 2 x + 1 -\left( x + 19 \right)\)

\(\displaystyle 0 > x^{2} + x - 18\)

Now we can use the quadratic formula.

Recall the quadratic formula.

\(\displaystyle \frac{ -b \pm \sqrt{ b ^2 -4 \cdot a \cdot c }}{ 2 \cdot a }\)

Where \(\displaystyle \uptext{a}\), \(\displaystyle \uptext{b}\), and \(\displaystyle \uptext{c}\), correspond to coefficients in the quadratic equation.

\(\displaystyle a x^{2} + b x + c = 0\)

In this case \(\displaystyle a = 1\) , \(\displaystyle b = 1\) , and \(\displaystyle c = -18\).

Now plug these values into the quadratic equation, and we get.

\(\displaystyle x = 3.8\)

\(\displaystyle x = -4.8\)

Now since we are dealing with an inequality, we put the least value on the left side, and the greatest value on the right. It will look like the following.

\(\displaystyle -4.8 < x < 3.8\)

To find the difference of two polynomials first set up the operation and identify the like terms.

\(\displaystyle \\A=x^2+2 \\B=4x^2-1\)

\(\displaystyle A-B= (x^2+2)-(4x^2-1)\)

The like terms in these polynomials are the squared variable and the constant terms.

Remember to distribute the negative sign through to all terms in the second polynomial.

\(\displaystyle A-B={\color{Red} x^2}{\color{Blue} +2}{\color{Red} -4x^2}{\color{Blue} --1}\)

\(\displaystyle \\x^2-4x^2=-3x^2 \\2-(-1)=3\)

Therefore, the sum of these polynomials is,

\(\displaystyle -3x^2+3\)

To find the sum of two polynomials first set up the operation and identify the like terms.

\(\displaystyle \\A=x^2+2 \\B=4x^2-1\)

\(\displaystyle A+B= (x^2+2)+(4x^2-1)\)

The like terms in these polynomials are the squared variable and the constant terms.

\(\displaystyle A+B={\color{Red} x^2}{\color{Blue} +2}{\color{Red} +4x^2}{\color{Blue} -1}\)

\(\displaystyle \\x^2+4x^2=5x^2 \\2+(-1)=1\)

Therefore, the sum of these polynomials is,

\(\displaystyle 5x^2+1\)

To find the difference of two polynomials first set up the operation and identify the like terms.

\(\displaystyle \\A=x^2+2 \\B=4x^2-1\)

\(\displaystyle B-A= (4x^2-1)-(x^2+2)\)

The like terms in these polynomials are the squared variable and the constant terms.

Remember to distribute the negative sign through to all terms in the second polynomial.

\(\displaystyle B-A={\color{Red} 4x^2}{\color{Blue} -1}{\color{Red} -x^2}{\color{Blue} -2}\)

\(\displaystyle \\4x^2-x^2=3x^2 \\-1-2=-3\)

Therefore, the sum of these polynomials is,

\(\displaystyle 3x^2-3\)

To find the difference of two polynomials first set up the operation and identify the like terms.

\(\displaystyle \\A=3x^3+x+2 \\B=4x^3-2x-1\)

\(\displaystyle A-B= (3x^3+x+2)-(4x^3-2x-1)\)

The like terms in these polynomials are the squared variable, the single variable, and the constant terms.

Remember to distribute the negative sign to all terms within the parentheses.

\(\displaystyle A-B={\color{Red} 3x^3}{\color{Blue} +x}+2{\color{Red} -4x^3}{\color{Blue} +2x}+1\)

\(\displaystyle \\3x^3-4x^3=-x^3 \\x+2x=3x \\2+1=3\)

Therefore, the sum of these polynomials is,

\(\displaystyle -x^3+3x+3\)

To find the sum of two polynomials first set up the operation and identify the like terms.

\(\displaystyle \\A=2x^2+2 \\B=4x^2-2\)

\(\displaystyle A+B= (2x^2+2)+(4x^2-2)\)

The like terms in these polynomials are the squared variable and the constant terms.

\(\displaystyle A+B={\color{Red} 2x^2}{\color{Blue} +2}{\color{Red} +4x^2}{\color{Blue} -2}\)

\(\displaystyle \\2x^2+4x^2=6x^2 \\2+(-2)=0\)

Therefore, the sum of these polynomials is,

\(\displaystyle 6x^2\)

To find the difference of two polynomials first set up the operation and identify the like terms.

\(\displaystyle \\A=2x^2+2 \\B=4x^2-2\)

\(\displaystyle A-B= (2x^2+2)-(4x^2-2)\)

The like terms in these polynomials are the squared variable and the constant terms.

Remember to distribute the negative sign to all terms in the second polynomial.

\(\displaystyle A-B={\color{Red} 2x^2}{\color{Blue} +2}{\color{Red} -4x^2}{\color{Blue} +2}\)

\(\displaystyle \\2x^2-4x^2=-2x^2 \\2+2=4\)

Therefore, the sum of these polynomials is,

\(\displaystyle -2x^2+4\)

To find the difference of two polynomials first set up the operation and identify the like terms.

\(\displaystyle \\A=2x^2+2 \\B=4x^2-2\)

\(\displaystyle B-A= (4x^2-2)-(2x^2+2)\)

The like terms in these polynomials are the squared variable and the constant terms.

Remember to distribute the negative sign to all terms in the second polynomial.

\(\displaystyle B-A= {\color{Red} 4x^2}-2{\color{Red} -2x^2}-2\)

\(\displaystyle \\4x^2-2x^2=2x^2 \\-2-2=-4\)

Therefore, the sum of these polynomials is,

\(\displaystyle 2x^2-4\)

To find the sum of two polynomials first set up the operation and identify the like terms.

\(\displaystyle \\A=x^2 \\B=4x^2-1\)

\(\displaystyle A+B= (x^2)+(4x^2-1)\)

The like terms in these polynomials are the squared variable.

\(\displaystyle A+B={\color{Red} x^2}{\color{Red} +4x^2}-1\)

\(\displaystyle \\x^2+4x^2=5x^2 \\0+(-1)=-1\)

Therefore, the sum of these polynomials is,

\(\displaystyle 5x^2-1\)

To find the product of two polynomials first set up the operation.

\(\displaystyle \\A=x^2+1 \\B=4x^2-1\)

\(\displaystyle AB=(x^2+1)(4x^2-1)\)

Now, multiply each term from the first polynomial with each term in the second polynomial.

Remember the rules of exponents. When like base variables are multiplied together their exponents are added together.

\(\displaystyle \\x^2\cdot 4x^2=4x^{2+2}=4x^4 \\x^2\cdot -1=-x^2 \\1\cdot 4x^2=4x^2 \\1\cdot -1=-1\)

Therefore, the product of these polynomials is,

\(\displaystyle 4x^4-x^2+4x^2-1\)

Combine like terms to arrive at the final answer.

 \(\displaystyle 4x^4+3x^2-1\)