To solve the equation [tex]\(-7x - 60 = x^2 + 10x\)[/tex] and rewrite it in factorized form, we can proceed as follows:
1. Rewrite the equation in standard form:
[tex]\[-7x - 60 = x^2 + 10x\][/tex]
2. Move all terms to one side to set the equation to zero:
[tex]\[
x^2 + 10x + 7x + 60 = 0
\][/tex]
3. Simplify the equation:
[tex]\[
x^2 + 17x + 60 = 0
\][/tex]
4. To find the equivalent equation in factorized form [tex]\((x + a)(x + b) = 0\)[/tex], we need to determine values for [tex]\(a\)[/tex] and [tex]\(b\)[/tex] such that the product [tex]\(a \cdot b = 60\)[/tex] and the sum [tex]\(a + b = 17\)[/tex].
After solving, we find:
[tex]\[a = -12\][/tex]
[tex]\[b = -5\][/tex]
5. Construct the factorized form of the equation:
[tex]\[
(x + a)(x + b) = (x - 12)(x - 5) = 0
\][/tex]
Therefore, the complete factorized equation is:
[tex]\[
(x - 12)(x - 5) = 0
\][/tex]
