The problem to solve is:

(x+20)(x-12)(x+4)>0 but I can't figure out how to do this right so I left off the >0 and tried to solve the equation.

x+20 evaluates to x+20
x-12 evaluates to x-12

Multiplying x+20 by x-12 is a classic Algebra problem. Here, you are trying
to multiply two binomials together (two expressions that each contain two terms).
Your book might call this finding the "Product of Two Binomials".

To work this problem, we'll use the "F.O.I.L." method. F.O.I.L. stands
for First, Outer, Inner, Last.

First, we'll multiply the two First terms, the x and x together.

Multiply x and x

Multiply the x and x

Multiply x and x

Combine the x and x by adding the exponents, and keeping the x, to get

The answer is



x × x =

Second, we'll multiply the two Outer terms, the x and -12 together.

Multiply x and -12

Multiply x and 1

The x just gets copied along.

The answer is x

x

x × -12 = -12x

Third, we'll multiply the two Inner terms, the 20 and x together.

Multiply 20 and x

Multiply 1 and x

The x just gets copied along.

x

20 × x = 20x

-12x combines with 20x to give 8x

Lastly, we'll multiply the two Last terms, the 20 and -12 together.

Multiply 20 and -12

1

20 × -12 = -240

(x+20)*(x-12) evaluates to

x+4 evaluates to x+4

Multiply by x+4

we multiply by each term in x+4 term by term.

This is the distributive property of multiplication.

Multiply and x

Multiply the and x

Multiply and x

Combine the and x by adding the exponents, and keeping the x, to get

The answer is



× x =

Multiply and 4

Multiply and 1

The just gets copied along.

The answer is



× 4 =

Multiply 8x and x

Multiply the x and x

Multiply x and x

Combine the x and x by adding the exponents, and keeping the x, to get

The answer is



8x × x =

combines with to give

Multiply 8x and 4

Multiply x and 1

The x just gets copied along.

The answer is x

x

8x × 4 = 32x

Multiply -240 and x

Multiply 1 and x

The x just gets copied along.

x

-240 × x = -240x

32x combines with -240x to give -208x

Multiply -240 and 4

1

-240 × 4 = -960

(x+20)*(x-12)*(x+4) evaluates to

x^3+12x^2-208x-960?

I used to teach a rather simple way to solve inequalities if they are factored.

in your case (x+20)(x-12)(x+4)>0

so we have 3 "critical" values
x = -20
x = -4
x = 12

marks these values on a number line, giving you 4 regions of numbers
1. for x<-20, pick any number in that region, work out the multiplication only considering the sign, not the actual answer
e.g. x = -30
(-)(-)(-) < 0 , no good
2. let x = -5 , between -20 and -4
(+)(-)(-) > 0 , good one
3. let x = 0 , between -4 and 12
(+)(-)(+) < 0 , no good
4. let x= 20 , greater than 12
(+)(+)(+) > 0 , good

so the result is positive between -20 and 4 as well as greater than 12

-20 < x < -4 OR x > 12

this method also works if you have division mixed in as well.

Of course if you know what the graph looks like, in this case a typical cubic with a + x^3 term, and you know the x-intercepts, you can just state the answer.

Thank you Reiny

The expression (x+20)(x-12)(x+4) evaluates to x^3 + 12x^2 - 208x - 960.

Now, to solve the inequality (x+20)(x-12)(x+4) > 0, we need to find the intervals where the expression is positive.

To do this, we can use the concept of interval notation and the sign of each factor.

1. Consider the factor (x + 20):

When x < -20, (x + 20) < 0
When x > -20, (x + 20) > 0

2. Consider the factor (x - 12):

When x < 12, (x - 12) < 0
When x > 12, (x - 12) > 0

3. Consider the factor (x + 4):

When x < -4, (x + 4) < 0
When x > -4, (x + 4) > 0

Now, we can combine the signs of each factor to determine the intervals where the expression is positive.

- When (x + 20) > 0, (x - 12) > 0, and (x + 4) > 0, we have x > -4 and x > 12.

So, the expression is positive in the interval (12, ∞).

- When (x + 20) < 0, (x - 12) < 0, and (x + 4) > 0, we have x < -4 and x < 12.

So, the expression is positive in the interval (-∞, -20).

Thus, the solution to the inequality (x+20)(x-12)(x+4) > 0 is x ∈ (-∞, -20) ∪ (12, ∞).

The problem you started with, (x+20)(x-12)(x+4), is actually an inequality, not an equation. The >0 at the end means that we are looking for values of x that make the expression greater than zero. To solve this inequality, we need to find the values of x that make the expression positive.

To do this, we can use a sign chart. First, let's find the x-values where the expression equals zero. Setting each factor equal to zero, we have x+20=0, x-12=0, and x+4=0. Solving these equations, we get x=-20, x=12, and x=-4.

Next, we select a test value for each of the resulting intervals (-∞, -20), (-20, -4), (-4, 12), and (12, +∞). A good test value for each interval is the number closest to the interval but not in it. For example, for the interval (-∞, -20), we can choose x=-21 as the test value.

Now we plug in the test values into the original inequality, (x+20)(x-12)(x+4)>0, and determine whether the expression is positive or negative. If it is positive, we mark that interval on the sign chart with a + sign, and if it is negative, we mark it with a - sign.

Using the test values, we find:

For the interval (-∞, -20): When x=-21, the expression is positive (+)(-)(-)>0, so we mark this interval with a +.

For the interval (-20, -4): When x=-10, the expression is positive (+)(-)(+)>0, so we mark this interval with a +.

For the interval (-4, 12): When x=0, the expression is negative (+)(-)(+)<0, so we mark this interval with a -.

For the interval (12, +∞): When x=13, the expression is positive (+)(+)(+)>0, so we mark this interval with a +.

Now we can see from the sign chart that the expression (x+20)(x-12)(x+4)>0 when x is in the intervals (-∞, -20) and (-4, 12). Therefore, the solution to the inequality is x∈(-∞, -20) U (-4, 12).