Question

express your answer as a polynomial in standard form.
$f(x) = -3x + 5$
$g(x) = x^2 + 4x - 15$
find: $g(f(x))$

Explanation:

Step1: Substitute \( f(x) \) into \( g(x) \)

We know \( f(x) = -3x + 5 \) and \( g(x)=x^{2}+4x - 15 \). So we substitute \( -3x + 5 \) for \( x \) in \( g(x) \), getting \( g(f(x))=(-3x + 5)^{2}+4(-3x + 5)-15 \).

Step2: Expand \( (-3x + 5)^{2} \)

Using the formula \( (a + b)^{2}=a^{2}+2ab + b^{2} \), where \( a=-3x \) and \( b = 5 \), we have \( (-3x)^{2}+2\times(-3x)\times5+5^{2}=9x^{2}-30x + 25 \).

Step3: Expand \( 4(-3x + 5) \)

Using the distributive property \( a(b + c)=ab+ac \), we get \( 4\times(-3x)+4\times5=-12x + 20 \).

Step4: Combine all terms

\( g(f(x))=9x^{2}-30x + 25-12x + 20-15 \). Now combine like - terms:

• For the \( x^{2} \) term: \( 9x^{2} \)
• For the \( x \) terms: \( -30x-12x=-42x \)
• For the constant terms: \( 25 + 20-15 = 30 \)

Answer:

\( g(f(x))=9x^{2}-42x + 30 \)