Question

question
express your answer as a polynomial in standard form.
$f(x) = 2x^2 - 5x + 15$
$g(x) = -2x - 2$
find: $f(g(x))$
answer attempt 1 out of 5

Explanation:

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

We know \( f(x) = 2x^2 - 5x + 15 \) and \( g(x) = -2x - 2 \). So we substitute \( x = g(x) = -2x - 2 \) into \( f(x) \), getting \( f(g(x)) = 2(-2x - 2)^2 - 5(-2x - 2) + 15 \).

Step2: Expand \( (-2x - 2)^2 \)

Using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \), where \( a = -2x \) and \( b = -2 \), we have \( (-2x - 2)^2 = (-2x)^2 + 2(-2x)(-2) + (-2)^2 = 4x^2 + 8x + 4 \). Then \( 2(-2x - 2)^2 = 2(4x^2 + 8x + 4) = 8x^2 + 16x + 8 \).

Step3: Expand \( -5(-2x - 2) \)

Using the distributive property, \( -5(-2x - 2) = 10x + 10 \).

Step4: Combine all terms

Now we substitute the expanded forms back into \( f(g(x)) \): \( f(g(x)) = 8x^2 + 16x + 8 + 10x + 10 + 15 \). Combine like terms: \( 8x^2 + (16x + 10x) + (8 + 10 + 15) = 8x^2 + 26x + 33 \).

Answer:

\( 8x^2 + 26x + 33 \)