The first two factors are the product of the sum and the difference of the same two terms, so we can use the difference of squares:

$\displaystyle (x - 4)(x+ 4) (x -7)$

$\displaystyle = (x^{2} - 4^{2}) (x -7)$

$\displaystyle = (x^{2} - 16) (x -7)$

Now use the FOIL method:

$\displaystyle = x^{2}\cdot x - x^{2} \cdot 7 - 16 \cdot x + 16 \cdot7$

$\displaystyle = x^{3} - 7x^{2} - 16x + 112$

$\displaystyle y^2z^3y^4z^5y^6$

Rearrange the expression so that the $\displaystyle \small y$ and $\displaystyle \small z$ variables of different powers are right next to each other.

$\displaystyle y^2y^4y^6z^3z^5$

When multiplying the same variable with different exponents, it is the same as adding the exponents: $\displaystyle a^ba^c=a^{b+c}$. Taking advantage of this rule, the problem can be rewritten.

$\displaystyle y^{2+4+6}z^{3+5}$

$\displaystyle y^{6+6}z^8$

$\displaystyle y^{12}z^8$

Rewrite each radical as an exponential expression, apply the product of powers property, and rewrite the product as a radical:

$\displaystyle \sqrt[4]{x} \cdot \sqrt[3]{x}$

$\displaystyle =x^{ \frac{1}{4}} \cdot x^{ \frac{1}{3}}$

$\displaystyle =x^{ \frac{1}{4}+ \frac{1}{3}}$

$\displaystyle =x^{ \frac{3}{12}+ \frac{4}{12}}$

$\displaystyle =x^{ \frac{7}{12}}$

$\displaystyle =\sqrt[12]{x ^{7}}$

This set is out of order, and it may be best to reorganize the terms.  To solve, this is very similar to the FOIL method. 

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

Follow the procedure to distribute each term.

$\displaystyle (a+b+c)(e+f+g)$

$\displaystyle = (ae+af+ag)+(be+bf+bg)+(ce+cf+cg)$

$\displaystyle =a(e+f+g)+b(e+f+g)+c(e+f+g)$

Follow suit to solve the problem.

$\displaystyle (-x^2+x-2)(3x^2-x+4)$

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

$\displaystyle =(-3x^4+x^3-4x^2)+(3x^3-x^2+4x)-6x^2+2x-8$

Combine like terms.

The correct answer is:  $\displaystyle -3x^4+4x^3-11x^2+6x-8$

The question is asking for the simplified version of this expression:

$\displaystyle \frac{3a^2b^4c}{4ab^2c^3} X \frac{12a^5b^2c}{3a^2bc^2}$

First, combine like terms in the numerator and denominator according to multiplication and exponent rules. When you are multiplying like bases together you add their exponents.

$\displaystyle \frac{3*12*a^2*a^5*b^4*b^2*c*c}{4*3*a*a^2*b^2*b*c^3*c^2}$

When you divide like bases you subtract the bottom exponent from the top exponent.

$\displaystyle \frac{36a^7b^6c^2}{12a^3b^3c^5} = (36/12)a^{7-3}b^{6-3}c^{2-5}$

Finally, simplify the expression by cancelling out terms using GCFs.

$\displaystyle 3a^{4}b^{3}c^{-3}$

$\displaystyle \frac{3a^4b^3}{c^3}$

The key to multiplying polynomials is to keep track of signs and be absolutely sure that you multiply every term in one polynomial (choose the first one) by every term in the other polynomial. For example if polynomial A has 3 terms and polynomial B has two, then then start with the first term in polynomial A and multiply it by both terms in polynomial B. Then choose the second term in polynomial A and repeat. Continue like so until you have gone through all iterations.

First term:

$\displaystyle ({\color{Red} 4x^3}+2x^2-8)({\color{Red} x-\frac{1}{4}})$

$\displaystyle 4x^4-x^3$

Second term:

$\displaystyle (4x^3+{\color{Green} 2x^2}-8)({\color{Green} x-\frac{1}{4}})$

$\displaystyle 4x^4-x^3+2x^3-\frac{1}{2}x^2$

Third term:

$\displaystyle (4x^3+2x^2-{\color{DarkBlue} 8})({\color{DarkBlue} x-\frac{1}{4}})$

$\displaystyle 4x^4-x^3+2x^3-\frac{1}{2}x^2-8x+2$

Now combine terms where able for the final answer:

$\displaystyle 4x^4+x^3-\frac{1}{2}x^2-8x+2$

Multiply each term of the first polynomial with all the terms of the second polynomial.  Follow the signs of the first polynomial.

$\displaystyle x^2(x^2+x+3) =x^4+x^3+3x^2$

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

$\displaystyle -3(x^2+x+3)= -3x^2-3x-9$

Add and combine like terms.

$\displaystyle x^4+x^3+3x^2+(-x^3-x^2-3x)+( -3x^2-3x-9)$

The answer is:  $\displaystyle x^4-x^2-6x-9$

The first step is to get everything out of parentheses to combine like terms. Since the polynomials are being subtracted, the sign of everything in the second polynomial will be flipped. You can think of this as a $\displaystyle -1$ being distributed across the polynomial:

$\displaystyle (14a^2 + 2a - 5) - (10a^2 - 3a + 8)$

$\displaystyle =(14a^2 + 2a - 5) + -1(10a^2 - 3a + 8)$

$\displaystyle =14a^2 + 2a - 5 - 10a^2 + 3a - 8$

Now combine like terms:

$\displaystyle 14a^2 + 2a - 5 - 10a^2 + 3a - 8$

$\displaystyle =4a^2 + 2a - 5 + 3a - 8$

$\displaystyle =4a^2 + 5a - 5 - 8$

$\displaystyle =4a^2 + 5a - 13$

Don't be scared by complex terms! First, check to see if the variables are alike. If they match perfectly, we can add and subtract their coefficients just like we could if the expression was $\displaystyle 3x - 3x$.

Remember, a variable is always a variable, no matter how complex! In this problem, the terms match! So we just subtract the coefficients of the matching terms and we get our answer:

$\displaystyle 3 x^2 z^4+7 y^3 z^3$

In simplifying this expression, be mindful of the order of operations (parenthical, division/multiplication, addition/subtraction).  

$\displaystyle 5x^{^{3}} - 14x^{2} - 10x + 3 - (4x+3) * x^{2} - (8-10x)$

Since operations invlovling parentheses occur first, distribute the factors into the parenthetical binomials. Note that the $\displaystyle x^2$ outside the first parenthetical binomial is treated as $\displaystyle -x^2$ since the parenthetical has a negative (minus) sign in front of it. Similarly, multiply the members of the expression in the second parenthetical by $\displaystyle -1$ because of the negative (minus) sign in front of it. Distributing these factors results in the following polynomial.

$\displaystyle 5x^3 - 14x^2 - 10x + 3 - 4x^3 - 3x^2 - 8 + 10x$

Now like terms can be added and subtracted. Arranging the members of the polynomial into groups of like terms can help with this. Be sure to retain any negative signs when rearranging the terms.

$\displaystyle 5x^3 - 4x^3 -14x^2 - 3x^2 - 10x + 10x + 3 - 8$

Adding and subtracting these terms results in the simplified expression below.

$\displaystyle x^3 - 17x^2 -5$