Question

1. f(x)= (x + 7)(x - 3) (-x - 7)(x - 3) (table with x, -7, x, -3 and cells -1x², -7x, 3x, 21; and -1x² -4x+... -48+12)

Explanation:

Step1: Expand the product \((x + 7)(x - 3)\)

Using the distributive property (FOIL method): \( (a + b)(c + d)=ac+ad+bc+bd \), here \( a = x \), \( b = 7 \), \( c = x \), \( d=-3 \).
So, \( (x + 7)(x - 3)=x\times x+x\times(-3)+7\times x + 7\times(-3) \)

Step2: Simplify each term

\( x\times x=x^{2} \), \( x\times(-3)=-3x \), \( 7\times x = 7x \), \( 7\times(-3)=-21 \)
Then combine like terms: \( x^{2}-3x + 7x-21=x^{2}+4x-21 \)
But there is a negative sign in front of \((x + 7)(x - 3)\) in \( f(x)=-(x + 7)(x - 3) \), so we apply the negative sign:
\( f(x)=-(x^{2}+4x - 21)=-x^{2}-4x + 21 \) (which matches the last line in the given work: \(-1x^{2}-4x+\dots\) with the constant term likely 21)

Answer:

If we are expanding \( f(x)=-(x + 7)(x - 3) \), the expanded form is \( \boldsymbol{-x^{2}-4x + 21} \)