g(x) = ( x / (x+3) ) + ( 3(2x+1) / (x^2 + x - 6) ). Show that this can be simplified to: g(x) = (x+1) / (x-2).
Step 1: The denominator of the right-hand fraction is quadratic, so we can factorise this to (x+3)(x-2). This looks similar to the denominator of the left-hand fraction, suggesting we can combine the two. Step 2: To make both denominators equal, multiply the left-hand fraction by (x-2)/(x-2). This is the same as multiplying by 1, so does not change anything. Step 3: The two fractions can now be combined into a single fraction: [ x(x-2) + 3(2x+1) ] / [ (x+3) (x-2) ]. By expanding the top line further, we obtain [ x^2 + 4x + 3 ] / [ (x+3) (x-2) ]. Step 4: The numerator of this fraction is quadratic, so just as in step 1, we can factorise this to [ (x+3) (x+1) ] / [ (x+3) (x-2) ]. Step 5: The (x+3) terms on the top and bottom both cancel out, leaving g(x) = (x+1) / (x-2).
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