To find the quadratic equation equivalent to [tex]\((x + 2)^2 + 5(x + 2) - 6 = 0\)[/tex], we will introduce a new variable [tex]\(u = (x + 2)\)[/tex]. This substitution will simplify the original equation. We can then work out the steps to transform the equation to its standard quadratic form.
1. Substitute [tex]\(u\)[/tex] for [tex]\((x+2)\)[/tex]:
[tex]\[ (u)^2 + 5(u) - 6 = 0 \][/tex]
2. Expand the equation:
[tex]\[ u^2 + 5u - 6 = 0 \][/tex]
3. Simplify:
This is already simplified in the standard quadratic form [tex]\(au^2 + bu + c = 0\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = 9\)[/tex], and [tex]\(c = -6\)[/tex].
Given the above manipulations and evaluations, the equivalent quadratic equation in terms of [tex]\(u\)[/tex] is:
[tex]\[ u^2 + 9u + 8 = 0 \][/tex]
Thus, the correct quadratic equation equivalent to the given expression is:
[tex]\[ u^2 + 9u + 8 = 0 \][/tex]
