To factor the quadratic expression completely, we start with the quadratic expression:
[tex]\[ -7x^2 - 24x - 9 \][/tex]
We will factor this expression into the product of two binomials. The factored form of the quadratic expression will look like:
[tex]\[ -(x + a)(7x + b) \][/tex]
where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are constants we need to determine.
By comparing the given quadratic expression to our proposed factored form, we can see that the leading coefficient [tex]\(-7\)[/tex] and the constant term [tex]\(-9\)[/tex] must factor into the terms of the binomials.
To find [tex]\(a\)[/tex] and [tex]\(b\)[/tex], we need to satisfy the following conditions:
1. The product of the constants [tex]\(-(x+a)(7x+b)\)[/tex] should give [tex]\(-7x^2 - 24x - 9\)[/tex].
2. The product of the constants inside the parentheses multiplied by [tex]\(-7\)[/tex] should return the original middle term and the constant term.
Using these criteria, we find:
[tex]\[
-(x + 3)(7x + 3)
\][/tex]
If we expand this factored form, we get back the original quadratic expression:
[tex]\[
-(x + 3)(7x + 3) = -(x \cdot 7x + x \cdot 3 + 3 \cdot 7x + 3 \cdot 3)
= -(7x^2 + 3x + 21x + 9)
= -7x^2 - 24x - 9
\][/tex]
Thus, the completely factored form of the given quadratic expression is:
[tex]\[
-(x + 3)(7x + 3)
\][/tex]
So, the answer is:
[tex]\[
-7x^2 - 24x - 9 = -(x + 3)(7x + 3)
\][/tex]
