Answer:
see explanation
Step-by-step explanation:
(a) (i)
x² - 9 ← is a difference of squares and factors as
(x - 3)(x + 3)
(ii)
given x² + x - 12
Consider the factors of the constant term (- 12) which sum to give the coefficient of the x- term (+ 1)
The factors are + 4 and - 3, since
4 × - 3 = - 12 and 4 - 3 = + 1, hence
x² + x - 12 = (x + 4)(x - 3)
(iii)
Express the numerator/ denominator in factored form
[tex]\frac{x^2-9}{x^2+x-12}[/tex]
= [tex]\frac{(x-3)(x+3)}{(x+4)(x-3)}[/tex]
Cancel the (x - 3) factor on the numerator/denominator, leaving
= [tex]\frac{x+3}{x+4}[/tex] with restriction x ≠ - 4
(b)
Expand the second pair of factors using FOIL
= (x + 2)(x² + 5x - 3x - 15)
= (x + 2)(x² + 2x - 15)
Multiply each term in the second factor by each term in the first factor
x(x² + 2x - 15) + 2(x² + 2x - 15) ← distribute both parenthesis
= x³ + 2x² - 15x + 2x² + 4x - 30 ← collect like terms
= x³ + 4x² - 11x - 30
Compare like terms with x³ + ax² - 11x + b
ax² with 4x² ⇒ a = 4
b with - 30 ⇒ b = - 30
