in your own words, explain each step below:\n-20x³ - 80x² - 12x - 48\n-4(5x³ + 20x² + 3x + 12)\n5x³ 20x²\n3x…

Brief Explanations:

Step 1:

Factor out the greatest common factor (-4) from each term of the polynomial (-20x^{3}-80x^{2}-12x - 48). Using the distributive property (ab+ac=a(b + c)), where (a=-4), (b = 5x^{3}+20x^{2}+3x + 12).

Step 2:

Group the terms inside the parentheses ((5x^{3}+20x^{2}+3x + 12)) into two - binomial groups ((5x^{3}+20x^{2})) and ((3x + 12)) for further factoring.

Step 3:

Factor out the common factors from each binomial group. From (5x^{3}+20x^{2}), factor out (5x^{2}) (since (5x^{3}+20x^{2}=5x^{2}(x + 4))), and from (3x + 12), factor out (3) (since (3x+12 = 3(x + 4))). The expression inside the parentheses can be rewritten as (5x^{2}(x + 4)+3(x + 4)).

Step 4:

Notice that ((x + 4)) is a common factor in the expression (5x^{2}(x + 4)+3(x + 4)).

Step 5:

Factor out the common binomial factor ((x + 4)) from (5x^{2}(x + 4)+3(x + 4)) using the distributive property (ab+ac=a(b + c)) where (a=(x + 4)), (b = 5x^{2}), and (c = 3). So, (5x^{2}(x + 4)+3(x + 4)=(5x^{2}+3)(x + 4)). Then, considering the factor (-4) from Step 1, the final factored form is (-4(5x^{2}+3)(x + 4)).

Answer:

• Step 1: Factor out the GCF (-4).
• Step 2: Group terms for factoring.
• Step 3: Factor out common factors from groups.
• Step 4: Identify the common binomial factor.
• Step 5: Factor out the common binomial factor.