The problem requires finding the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], given that the polynomial [tex]\(p(x)\)[/tex] equals the polynomial [tex]\(q(x)\)[/tex]. We are given the polynomials:
[tex]\[ p(x) = (a-2)x^2 + (b+1)x + (c+2) \][/tex]
[tex]\[ q(x) = 4x^2 + 5x + 7 \][/tex]
Since [tex]\(p(x) = q(x)\)[/tex], the coefficients of the corresponding powers of [tex]\(x\)[/tex] in both polynomials must be equal. Let's compare the coefficients term by term:
1. Coefficient of [tex]\(x^2\)[/tex]:
[tex]\[ (a-2)x^2 \quad \text{and} \quad 4x^2 \][/tex]
Equating the coefficients of [tex]\(x^2\)[/tex]:
[tex]\[ a - 2 = 4 \][/tex]
Solving for [tex]\(a\)[/tex]:
[tex]\[ a = 4 + 2 \][/tex]
[tex]\[ a = 6 \][/tex]
2. Coefficient of [tex]\(x\)[/tex]:
[tex]\[ (b+1)x \quad \text{and} \quad 5x \][/tex]
Equating the coefficients of [tex]\(x\)[/tex]:
[tex]\[ b + 1 = 5 \][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[ b = 5 - 1 \][/tex]
[tex]\[ b = 4 \][/tex]
3. Constant term:
[tex]\[ (c+2) \quad \text{and} \quad 7 \][/tex]
Equating the constant terms:
[tex]\[ c + 2 = 7 \][/tex]
Solving for [tex]\(c\)[/tex]:
[tex]\[ c = 7 - 2 \][/tex]
[tex]\[ c = 5 \][/tex]
Therefore, the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are:
[tex]\[ \boxed{a = 6, b = 4, c = 5} \][/tex]
