Answer:a = 1 and b = -12Step-by-step explanation:* Lets look to the graph to solve the problem- f(x) is a rational function with:# numerator ⇒ x² - 4x + 3# denominator ⇒ x² + ax + b- To find the values of a and b lets use the hole and the vertical asymptote in the graph of the function∵ There is a hole at x = 3- The hole appears when the numerator and the denominator have a common factor ∵ x = 3 ⇒ subtract 3 from both sides∴ x - 3 = 0∴ The numerator and the denominator have a factor (x - 3)- The vertical asymptotes appear at the zeros of the denominator that means the values of x when the denominator = 0∵ There is a vertical asymptote at x = -4∵ x = -4 ⇒ add 4 for both sides∴ x + 4 = 0∴ The denominator has another factor (x + 4)- The denominator x² + ax + b has two factors (x - 3) and (x + 4) lets multiply them to find the value of a and b∵ (x - 3)(x + 4) = (x)(x) + (x)(4) + (-3)(x) +(-3)(4)∴ (x - 3)(x + 4) = x² + 4x - 3x - 12 = x² + x - 12∴ x² + ax + b = x² + x - 12- By comparing the two sides∴ a = 1 and b = -12