Alright, let's break down the given expression [tex]\( -3x^2 - 24x - 36 \)[/tex] step by step.
1. First, notice that each term in the expression contains a common factor of [tex]\(-3\)[/tex]. We can factor out [tex]\(-3\)[/tex]:
[tex]\[ -3(x^2 + 8x + 12) \][/tex]
2. Next, we need to factor the quadratic expression inside the parentheses: [tex]\( x^2 + 8x + 12 \)[/tex].
3. We look for two numbers that multiply to [tex]\(12\)[/tex] (the constant term) and add to [tex]\(8\)[/tex] (the coefficient of [tex]\(x\)[/tex]).
These numbers are [tex]\(6\)[/tex] and [tex]\(2\)[/tex] because:
[tex]\[ 6 \times 2 = 12 \][/tex]
[tex]\[ 6 + 2 = 8 \][/tex]
4. So, we can factor [tex]\( x^2 + 8x + 12 \)[/tex] as:
[tex]\[ (x + 6)(x + 2) \][/tex]
5. Finally, substituting this back into the expression we factored out:
[tex]\[ -3(x + 6)(x + 2) \][/tex]
Therefore, the expression [tex]\(-3x^2 - 24x - 36\)[/tex] is equivalent to:
[tex]\[ -3(x + 6)(x + 2) \][/tex]
So, choosing the correct elements from the drop-down menus, you should have:
- The first blank: [tex]\(-3\)[/tex]
- The second blank: [tex]\(6\)[/tex]
- The third blank: [tex]\(2\)[/tex]
Putting it all together, the equivalent expression is:
[tex]\[ \boxed{-3} (x + \boxed{6})(x + \boxed{2}) \][/tex]
